- Email: [email protected]
ROSFIT: An enzyme kinetics nonlinear regression curve fitting package for a microcomputer
COMPUTERS
AND
ROSFIT:
BIOMEDICAL
RESEARCH
15,
39-45 (1982)
An Enzyme Kinetics Nonlinear Regression Fitting Package for a Microcomputer
WILLIAM R. Gmco,**t
*Computer Roswell
ROGERL. PRIORE,* MOHESWAR SHARMA,t WALTER KORYTNYKt
Center and tDepartment Park Memorial institute,
of Experimental Buffalo, New
Curve
AND
Therapeutics, York 14263
Received July 6, 1981
A nonlinear regression curve fitting package has been specifically developed for enzyme kinetic analyses for use on the Hewlett-Packard HP-85 microcomputer. Data are entered in a conversational manner. Data can be changed, deleted or added, and data sets can be stored and retrieved from a magnetic tape cassette. Data can be fit to any of nine models: the Michaelis-Menten equation, substrate inhibition, random bi bi, ordered bi bi, ping pong bi bi, competitive inhibition, classical noncompetitive inhibition, modern noncompetitive inhibition, or uncompetitive inhibition. The printout for each model consists of several goodness-of-fit statistics, the parameter estimates with 95% confidence intervals, the variancecovariance, and correlation matrices, a residual analysis, and graphs. For example, for competitive inhibition the graphs provided are u vs [a, u vs [I], l/u vs l/[S], and l/u vs [a with up to four concentrations of the second compound plotted on each graph. The nonlinear regression algorithm in the package is that of Marquardt. The values determined by ROSFIT are essentially the same as those found using the BMDPAR, BMDP3R, and NONLIN programs on a Univac 90/60 mainframe computer. A typical run for competitive inhibition with 46 data points took a total time of about 18.5 min, not including data entry time, 14 min for the graph generation, 2 min for the three required iterations, and 2.5 min for miscellaneous operations.
For estimating enzyme kinetic parameters and for discriminating between rival kinetic models, in terms of accuracy and precision, the best methods involve the use of nonlinear regression curve fitting (Z-3). With the increased accessibility of digital computers, nonlinear curve fitting is becoming increasingly popular, although the most common approach to analyzing enzyme kinetic data is still the double reciprocal Lineweaver-Burk plot (4). The first published method for estimating K, and V,,, using nonlinear curve-fitting techniques was that by Wilkinson (5). Later Cleland (6-8) and Atkins and Nimmo (9) described nonlinear curve fitting in biochemistry and the statistical analysis of enzyme kinetic data in a detailed and understandable manner. There are many general nonlinear regression curve-fitting packages available for use on mainframe computers, e.g., BMDPAR (10) and MLAB (II). NONLIN (12) is commonly used by pharmacokineticists. In addition, there are 39 001~4809/82/010039-07$02.00/O Copyright @I 1982 by Academic Press, Inc. AU rights of reproduction in any form reserved.
GRECOETAL.
40
nonlinear curve-fitting packages available for implementation on mainframes, but written specifically for use in enzyme kinetics (7, 8, 13 -15). The use of these program packages for enzyme kinetic analyses is still limited. Possible reasons for this include the historical tradition and simplicity of linearized graphical plots, the lack of access of many investigators to a large mainframe computer, the cost and inconvenience of implementing a program package on a maini?ame other than that for which it was originally written; interface, phoneline, and computer time costs for a mainframe, and the time and effort needed to learn the rudiments of computer programming and computer systems operations. Possibly the answer to the above problems is to have programs for nonlinear curve fitting specifically tailored for enzyme kinetics and running on relatively inexpensive microcomputers located in the researcher’s laboratory. In the last several years microcomputers have become available at low cost, with adequate memory space, and which run quickly enough to make this idea possible. This paper reports on a program package for the analysis of enzyme kinetics; it is called ROSFIT and runs on a Hewlett-Packard HP-85 microcomputer. METHODS Computer Hardware. The HP-85 microcomputer (Hewlett-Packard, Corvallis, Ore.) has a keyboard, 5-in. video monitor, digital-tape cassette recorder, thermal matrix printer, and 16 kilobytes of memory housed in one compact unit. Although the computer is upgradable to 32 kbyte of memory, and it is possible to add minifloppy disk drives, a pen plotter, and other accessories, ROSFIT is written to run on the minimum configuration. Computer Software. ROSFIT is a set of 11 separate programs written in BASIC, a data entry program, a program for drawing graphs, and separate programs for fitting each of nine models: the Michaelis-Menten equation, u = VA/(K
El1
+ A);
substrate inhbition, u = VA/(K
PI
+ A + A2/KJ;
random bi bi, u = VAB/(K,,,Kb
+ K,,B + Kd
+ AB) ;
[31
ordered bi bi, u = VAB/(K,Kb
+ K,,A + All);
[41
+ K&
151
ping pong bi bi, u = VAB/(KaKb
competitive
inhibition,
+ All) ;
NONLINEAR
u = VA/[K(I uncompetitive
+ Z/K,> + A];
FOR MICROCOMPUTERS
41 C6J
inhibition,
u = VA/[K
+ A(1 + Z/K,)];
classical noncompetitive u = VA/[@1 and modem
REGRESSION
inhibition,
+ Z/Z&) + A( 1 + Z/KJ];
noncompetitive
u = VA/[K( 1 + Z/K,)
[71 PI
inhibition, + A( 1 + Z/Z&)].
c91
A complete explanation of the derivation of the equations, symbol notation,’ and recommendations for experimental design when using these models has been outlined by Cleland (8). The data-entry program allows data to be entered in a conversational manner, allows data to be changed, added, or deleted, and allows data sets to be stored or loaded from a digital cassette tape. One data set can contain up to 75 data points. Each of the model-fitting programs begins by generating initial estimates of the parameters by a weighted multiple linear regression analysis of the double reciprocal form of the appropriate equations. Using these initial estimates, the nonlinear regression routine which uses the Marquardt (16) algorithm as modified by Nash (27) iterates to the final parameter estimates. The data can be unweighted, weighted by the reciprocal of the calculated velocity, or weighted by the square of the reciprocal of the calculated velocity. (This last weighting factor is appropriate when the error in the velocity is proportional to the magnitude of the velocity (9).) The weights are normalized by making the sum of the weights equal to the number of data points. The printout includes a table of the raw data points, the final weighted sum of squares, the residual mean square, the variance-covariance and correlation matrices, the final parameters with their asymptotic standard errors, the 95% confidence limits for the parameters, and a table of the residuals. These statistics will give an indication of the reliability of the overall fit of the data to a particular model, and an indication of the reliability of each of the parameter estimates. A more complete explanation of the use of these statistics in interpreting the results can be found in Cleland (8), and in the documentation for ROSFIT. All of the nine model-fitting programs automatically call the graph generation 1 The symbols used are: K, substrate complex, depending V, the maximum velocity; A,
the Michaelis constant or the dissociation constant of the enzymeupon the equation and the particular assumptions in the derivation; the concentration of the first substrate; B, the concentration of the second substrate, I, the inhibitor concentration; Ku, the dissociation constant of the first substrateenzyme complex; K,, the Michaelis constant for substrate A, Kbr the Michaelis constant for substrate B; Kti, the dissociation constant of the enzyme-inhibitor complex; K,,, the dissociation constant of the enzyme-inhibitor-substrate complex; K,, the dissociation constant of both the enzyme-inhibitor and the enzyme-inhibitor-substrate complexes.
42
GRECO ET AL.
program with a “CHAIN” command. For the Michaelis-Menten equation, and for substrate inhibition, two graphs are generated: velocity vs substrate concentration and a double reciprocal plot. For random bi bi, ordered bi bi, or ping pong bi bi, four graphs are generated: velocity vs the concentration of the first substrate (A), velocity vs the concentration of the second substrate (B), l/v vs l/A, and l/u vs l/B. Competitive inhibition, classical noncompetitive inhibition, modern noncompetitive inhibition, or uncompetitive inhibition generate four graphs: u vs A, velocity vs inhibitor concentration (Z), I/v vs l/A, and l/v vs 1. Except for the graphs for the Michaelis-Menten equation, curves for up to four concentrations of the second compound will be plotted simultaneously on each of the graphs. Hexosaminidase Assay. The hexosaminidase assay (EC 3.2.1.52) was modified from (18). Briefly, the final reaction mixture of 0.2 ml contains 13.5 mM citrate, 23 mM phosphate (pH 4.5), 0.1% bovine serum albumin, the substrate (pnitrophenyl-N-acetyl-/3-D-glucosaminide), inhibitors, and enzyme. Controls contain no substrate. The reaction is carried out at 37°C for 30 min, and is terminated by adding 1 ml of 0.25 it4 glycine buffer, pH 9.8. The reaction is measured by an increase of the absorbance at 400 nm using a Gilford Model 300N spectrophotometer (Gilford Instrument Laboratories, Oberlin, Ohio). Hexosaminidase was purchased from Boehringer Mannheim Biochemicals, Indianapolis, Indiana, and p-nitrophenyl-l\r-acetyl-/3-D-glucosaminide from Sigma Chemical Company, St. Louis Missouri. The synthesis of the inhibitor, 2-acetamido-2,4-dideoxy-4-fluoro-D-galactono-1,5-lactone will be described in a paper under preparation.
RESULTS AND DISCUSSION
Both real data and contrived data were fit to the nine different models. The real data was from the inhibition of hexosaminidase by 2-acetamido-2,4dideoxy-Uluoro-D-galactono1,5-lactone, using p-nitrophenyl-N-acetyl-/3-Dglucosaminide as the substrate. The results from this experiment, comprising 46 data points, were fit to all four inhibition models. Sets of contrived data were fit to the other models. When fitting sets of data to any of the models without weighting, ROSFIT gave the same values to at least three significant figures for the parameters and the various statistics as did the commonly used mainframe statistical software packages BMDPAR, BMDP3R (ZO), and NONLIN (12). When sets of data were fit to the models with the iterative reweighting options, the values of the parameters usually agreed to only 1 or 2 significant figures. The complications of changing weighting factors at each iteration combined with somewhat different nonlinear regression algorithms and different convergence criteria in the various software packages probably led to these differences. The real inhibition data fit the unweighted competitive inhibition model best with K = 847 r 26.6 Z.L.Mand Ku = 27.2 r 0.876 PM. Figure 1 is a copy of the v vs A plot from the actual printout
NONLINEAR
REGRESSION
FOR MICROCOMPUTERS
!
43
113
0
1.86
2.12 CA3
3.18
4.25
. +
= .2
0
=0
=
5
FIG. 1. A copy of the actual printout from ROSFIT for the plot of velocity vs substrate concentration for the inhibition of hexosaminidase by 2-acetamido-2,Cdideoxy-Qfluoro-ogalactono- 1.5~lactone using p-nitrophenyl-N-acetyl-B-D-glucosaminide as the substrate. The units for velocity are change in absorbance per 30 min, and the units for substrate concentration A and inhibitor concentration Z are mM. The symbols used for the three concentrations of inhibitor are defined in the figure. The original printout has blue-dot matrix characters which do not photograph very well.
from this run. The quality of the printout is somewhat better than it appears in the figure; the blue dot matrix printing does not photograph well. The execution time of ROSFIT is adequate for enzyme kinetic applications. For example, excluding data entry, for the unweighted competitive inhibition example of Fig. 1, the total execution time was 18.5 min; 2 min for the 3 iterations, 14 min for the graph generation, and 2.5 min for miscellaneous operations. For a run with the Michaelis-Menten equation with 12 data points and the weighting factor of the reciprocal of the squared calculated velocity, the total execution time was about 4.5 min; 23 s for the 1 iteration, 2.5 min for the graph generation, and about 1.5 min for miscellaneous operations. The implementation of nonlinear regression algorithms on microcomputers and programmable calculators is relatively new, but expanding. Peck and Ban-et (19) adapted two previously written BASIC programs, one by Patlak and Pettigrew and one by Horowitz and Homer (20), to the Commodore PET microcomputer. Muri (21) has written a nonlinear regression program on a Texas Instrument TI-59 programmable calculator for fitting data to some simple pharmacokinetic models. Duggleby (22) has recently published a short general BASIC program for fitting data to two-parameter nonlinear models, which can be implemented on microcomputers. Koeppe and Hamann have also recently published a short general BASIC program for nonlinear curve fitting (23). We intend to update ROSFIT to include some more complicated enzyme kinetic models, and to include specific curve fitting programs for the analysis of data for ligand-macromolecule binding, radioimmunoassay, dose-response analysis, pharmacokinetics, transport studies, and survival analyses. ROSFIT and most future extensions of the package will consist of application-tailored programs. The major advantages of specific, application-tailored programs
44
GRECO ET AL.
over more general curve-fitting programs are that the user does not have to understand programming to use the program, the user does not have to do so much work with each run, and the graphical output can be optimized for each specific application. There were several other microcomputers considered for this project. They included the Apple II (Apple Computer, Cupertino, Calif.), Radio Shack TRS-80 (Tandy Corp., Fort Worth, Tex.), Texas Instrument 99/4 (Texas Instruments, Dallas, Tex.), and North Star Horizon (North Star Computers, Berkeley, Calif.). A system could have been configured around any of these micros instead of the HP-85 to complete this project. In fact, for the same price as that paid for the HP-85,” most of these systems would have included a minifloppy disk drive instead of a digital cassette tape recorder. But overall, the HP-85 was chosen because of its compactness, ease of programming, excellent documentation, good graphics, price, and miscellaneous features which make it suited for scientific purposes as opposed to business or hobby applications. Especially useful is the wide range of real numbers allowed with the HP-85. Real numbers are represented internally with 12 digits and a 3-digit exponent in the range -499 through 499. In summary, an enzyme kinetics curve-fitting package is now available for use on the HP-85 microcomputer. Advantages over conventional graphical analyses include increased accuracy and precision of parameter estimates, a savings of time, and an increased yield of useful information from experiments. Advantages over analyses using large mainframe computers include convenience, a possible savings of time if the mainframe is located outside of one’s laboratory, and graphics tailored specifically to a particular enzyme kinetic model. ROSFIT was written not just as an exercise to prove that sophisticated data analyses can be performed on a microcomputer norjust for data analyses in our laboratories. ROSFIT is intended for use by the biomedical community at large. Documentation on how to use the program and interpret the results is available. Instructions on how to receive a tape cassette containing ROSFIT and the documentation can be obtained by writing to the first author. ACKNOWLEDGMENTS We thank Dr. John Bertram and Dr. Jerry Roth for allowing the use of the microcomputers located in their laboratories. We thank Angela M. Di Loro for performing the hexosaminidase assays, and we thank Diana Fischer for running most of the analyses on the HP-85 and the Univac 90/60. This work was supported in part by Grants CA-16056 and CA-13038 from the U.S. Public Health Service. REFERENCES G. L., AND NIMMO, I. A. A comparison of seven methods for fitting the MichaelisMenten equation. Biochem. J. 149, 775 (1975).
1. ATKIN,
* The retail price of the HP-85 is $3250.
NONLINEAR
REGRESSION
FOR MICROCOMPUTERS
45
I. A., AND ATKINS, G. L. Methods for fitting equations with two or more non-linear parameters. Biochem. J. 157, 489 (1976). GRECO, W. R., AND HAKALA, M. T. Evaluation of methods for estimating the dissociation constant of tight binding enzyme inhibitors. J. Biol. Chem. 254, 12104 (1979). LINEWEAVER, H., AND Buax, D. The determination of enzyme dissociation constants. J. Amer. Chem. Sot. 56, 658 (1934). WILKINSON, G. N. Statistical estimations in enzyme kinetics. Biochem. J. 80, 324 (1961). CLELAND, W. W. The Kinetics of enzyme-catalyzed reactions with two or more substrates or products. Biochim. Biophys. Acta 67, 173 (1963). CLELAND, W. W. The statistical analysis of enzyme kinetic data. In “Advances in Enzymology” (F. F. Nord, Ed.), Vol. 29, pp. l-32. Wiley, New York, 1967. CLELAND, W. W. Statistical analysis of enzyme kinetic data. In “Methods in Enzymology” (D. L. Purich, Ed.), Vol. 63, pp. 103-138. Academic Press, New York, 1979. ATKINS, G. L., AND NIMMO, I. A. Current trends in the estimation of Michaelis-Menten parameters. Biochem. 104, 1 (1980). RALSTON, M. In “BMDP Biomedical Computer Programs” (W. J. Dixon and M. B. Brown, Eds.), pp. 484-498. Univ. of Calif. Press, Berkeley, 1977. KNOTT, G. D. MLAB-A mathematical modeling tool. Camp. Prog. Biomed. 10, 271 (1979). METZLER, C. M., ELFRING, G. L., AND MC EWEN, A. J. A package of computer programs for pharmacokinetic modeling. Biometrics 30, 562 (1974). KOHN, M. C., MENTEN, L. E., AND GARFINKEL, D. A convenient computer program for fitting enzymatic rate laws to steady-state data. Camp. Biomed. Res. 12, 461 (1979). HANSON, K. R., LING, R., AND HAVIR, E. A computer program for fitting data to the Michaelis-Menten equation. Biochem. Biophys. Res. Comm. 29 (1967). SIANO, D. B., ZYSKIRD, J. W., AND FROMM, H. J. A computer program for fitting and statistically analyzing initial rate data applied to bovine hexokinase type II isozyme. Arch.
2. NIMMO, 3. 4. 5. 6. 7. 8. 9.
10. 11. 12. 13. 14. 15.
Biochem.
Biophys.
170, 587 (1975).
16. MARQUARDT, D. W. An algorithm for least squares estimation of nonlinear parameters. J. Sot. Ind. Appl. Marh. 11, 431 (1963). 17. NASH, J. C. ‘Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation.” Wiley, New York, 1979. 18. VON FIGURA, K. Secretion of /3-Hexosaminidase by cultured human skin fibroblasts. Exp. Cell. Res. 111, 15 (1978). 19. PECK, C. C., AND BARRETT, B. B. Nonlinear least-squares regression program for microcomputers. .I. Pharmacokin. Biopharm. 7, 537 (1979). 20. HOROWITZ, D. L., AND HOMER, L. D. Analysis of biomedical data by time-sharing computers IV. Non-Linear regression analysis. Project No. MRO05: 200287, Rep. No. 25, Naval Med. Res. Inst. Nat. Naval Med. Ctr., Bethesda, Md. 20014, Feb. 26, 1970. 21. MUIR, K. T. Nonlinear least-squares regression analysis in pharmacokinetics: Application of a programmable calculator in model parameter estimation. Camp. Biomed. Res. 13, 307’(1980). 22. DUGGLEBY, R. G. A Nonlinear regression program for small computers. Anal. Biochem. 110,9 (1981). 23. KOEPPE, P., AND I&MANN, C. A program for non-linear regression analysis to be used on desk-top computers. Camp. Prog. Biomed. 12, 121 (1980).
AND
ROSFIT:
BIOMEDICAL
RESEARCH
15,
39-45 (1982)
An Enzyme Kinetics Nonlinear Regression Fitting Package for a Microcomputer
WILLIAM R. Gmco,**t
*Computer Roswell
ROGERL. PRIORE,* MOHESWAR SHARMA,t WALTER KORYTNYKt
Center and tDepartment Park Memorial institute,
of Experimental Buffalo, New
Curve
AND
Therapeutics, York 14263
Received July 6, 1981
A nonlinear regression curve fitting package has been specifically developed for enzyme kinetic analyses for use on the Hewlett-Packard HP-85 microcomputer. Data are entered in a conversational manner. Data can be changed, deleted or added, and data sets can be stored and retrieved from a magnetic tape cassette. Data can be fit to any of nine models: the Michaelis-Menten equation, substrate inhibition, random bi bi, ordered bi bi, ping pong bi bi, competitive inhibition, classical noncompetitive inhibition, modern noncompetitive inhibition, or uncompetitive inhibition. The printout for each model consists of several goodness-of-fit statistics, the parameter estimates with 95% confidence intervals, the variancecovariance, and correlation matrices, a residual analysis, and graphs. For example, for competitive inhibition the graphs provided are u vs [a, u vs [I], l/u vs l/[S], and l/u vs [a with up to four concentrations of the second compound plotted on each graph. The nonlinear regression algorithm in the package is that of Marquardt. The values determined by ROSFIT are essentially the same as those found using the BMDPAR, BMDP3R, and NONLIN programs on a Univac 90/60 mainframe computer. A typical run for competitive inhibition with 46 data points took a total time of about 18.5 min, not including data entry time, 14 min for the graph generation, 2 min for the three required iterations, and 2.5 min for miscellaneous operations.
For estimating enzyme kinetic parameters and for discriminating between rival kinetic models, in terms of accuracy and precision, the best methods involve the use of nonlinear regression curve fitting (Z-3). With the increased accessibility of digital computers, nonlinear curve fitting is becoming increasingly popular, although the most common approach to analyzing enzyme kinetic data is still the double reciprocal Lineweaver-Burk plot (4). The first published method for estimating K, and V,,, using nonlinear curve-fitting techniques was that by Wilkinson (5). Later Cleland (6-8) and Atkins and Nimmo (9) described nonlinear curve fitting in biochemistry and the statistical analysis of enzyme kinetic data in a detailed and understandable manner. There are many general nonlinear regression curve-fitting packages available for use on mainframe computers, e.g., BMDPAR (10) and MLAB (II). NONLIN (12) is commonly used by pharmacokineticists. In addition, there are 39 001~4809/82/010039-07$02.00/O Copyright @I 1982 by Academic Press, Inc. AU rights of reproduction in any form reserved.
GRECOETAL.
40
nonlinear curve-fitting packages available for implementation on mainframes, but written specifically for use in enzyme kinetics (7, 8, 13 -15). The use of these program packages for enzyme kinetic analyses is still limited. Possible reasons for this include the historical tradition and simplicity of linearized graphical plots, the lack of access of many investigators to a large mainframe computer, the cost and inconvenience of implementing a program package on a maini?ame other than that for which it was originally written; interface, phoneline, and computer time costs for a mainframe, and the time and effort needed to learn the rudiments of computer programming and computer systems operations. Possibly the answer to the above problems is to have programs for nonlinear curve fitting specifically tailored for enzyme kinetics and running on relatively inexpensive microcomputers located in the researcher’s laboratory. In the last several years microcomputers have become available at low cost, with adequate memory space, and which run quickly enough to make this idea possible. This paper reports on a program package for the analysis of enzyme kinetics; it is called ROSFIT and runs on a Hewlett-Packard HP-85 microcomputer. METHODS Computer Hardware. The HP-85 microcomputer (Hewlett-Packard, Corvallis, Ore.) has a keyboard, 5-in. video monitor, digital-tape cassette recorder, thermal matrix printer, and 16 kilobytes of memory housed in one compact unit. Although the computer is upgradable to 32 kbyte of memory, and it is possible to add minifloppy disk drives, a pen plotter, and other accessories, ROSFIT is written to run on the minimum configuration. Computer Software. ROSFIT is a set of 11 separate programs written in BASIC, a data entry program, a program for drawing graphs, and separate programs for fitting each of nine models: the Michaelis-Menten equation, u = VA/(K
El1
+ A);
substrate inhbition, u = VA/(K
PI
+ A + A2/KJ;
random bi bi, u = VAB/(K,,,Kb
+ K,,B + Kd
+ AB) ;
[31
ordered bi bi, u = VAB/(K,Kb
+ K,,A + All);
[41
+ K&
151
ping pong bi bi, u = VAB/(KaKb
competitive
inhibition,
+ All) ;
NONLINEAR
u = VA/[K(I uncompetitive
+ Z/K,> + A];
FOR MICROCOMPUTERS
41 C6J
inhibition,
u = VA/[K
+ A(1 + Z/K,)];
classical noncompetitive u = VA/[@1 and modem
REGRESSION
inhibition,
+ Z/Z&) + A( 1 + Z/KJ];
noncompetitive
u = VA/[K( 1 + Z/K,)
[71 PI
inhibition, + A( 1 + Z/Z&)].
c91
A complete explanation of the derivation of the equations, symbol notation,’ and recommendations for experimental design when using these models has been outlined by Cleland (8). The data-entry program allows data to be entered in a conversational manner, allows data to be changed, added, or deleted, and allows data sets to be stored or loaded from a digital cassette tape. One data set can contain up to 75 data points. Each of the model-fitting programs begins by generating initial estimates of the parameters by a weighted multiple linear regression analysis of the double reciprocal form of the appropriate equations. Using these initial estimates, the nonlinear regression routine which uses the Marquardt (16) algorithm as modified by Nash (27) iterates to the final parameter estimates. The data can be unweighted, weighted by the reciprocal of the calculated velocity, or weighted by the square of the reciprocal of the calculated velocity. (This last weighting factor is appropriate when the error in the velocity is proportional to the magnitude of the velocity (9).) The weights are normalized by making the sum of the weights equal to the number of data points. The printout includes a table of the raw data points, the final weighted sum of squares, the residual mean square, the variance-covariance and correlation matrices, the final parameters with their asymptotic standard errors, the 95% confidence limits for the parameters, and a table of the residuals. These statistics will give an indication of the reliability of the overall fit of the data to a particular model, and an indication of the reliability of each of the parameter estimates. A more complete explanation of the use of these statistics in interpreting the results can be found in Cleland (8), and in the documentation for ROSFIT. All of the nine model-fitting programs automatically call the graph generation 1 The symbols used are: K, substrate complex, depending V, the maximum velocity; A,
the Michaelis constant or the dissociation constant of the enzymeupon the equation and the particular assumptions in the derivation; the concentration of the first substrate; B, the concentration of the second substrate, I, the inhibitor concentration; Ku, the dissociation constant of the first substrateenzyme complex; K,, the Michaelis constant for substrate A, Kbr the Michaelis constant for substrate B; Kti, the dissociation constant of the enzyme-inhibitor complex; K,,, the dissociation constant of the enzyme-inhibitor-substrate complex; K,, the dissociation constant of both the enzyme-inhibitor and the enzyme-inhibitor-substrate complexes.
42
GRECO ET AL.
program with a “CHAIN” command. For the Michaelis-Menten equation, and for substrate inhibition, two graphs are generated: velocity vs substrate concentration and a double reciprocal plot. For random bi bi, ordered bi bi, or ping pong bi bi, four graphs are generated: velocity vs the concentration of the first substrate (A), velocity vs the concentration of the second substrate (B), l/v vs l/A, and l/u vs l/B. Competitive inhibition, classical noncompetitive inhibition, modern noncompetitive inhibition, or uncompetitive inhibition generate four graphs: u vs A, velocity vs inhibitor concentration (Z), I/v vs l/A, and l/v vs 1. Except for the graphs for the Michaelis-Menten equation, curves for up to four concentrations of the second compound will be plotted simultaneously on each of the graphs. Hexosaminidase Assay. The hexosaminidase assay (EC 3.2.1.52) was modified from (18). Briefly, the final reaction mixture of 0.2 ml contains 13.5 mM citrate, 23 mM phosphate (pH 4.5), 0.1% bovine serum albumin, the substrate (pnitrophenyl-N-acetyl-/3-D-glucosaminide), inhibitors, and enzyme. Controls contain no substrate. The reaction is carried out at 37°C for 30 min, and is terminated by adding 1 ml of 0.25 it4 glycine buffer, pH 9.8. The reaction is measured by an increase of the absorbance at 400 nm using a Gilford Model 300N spectrophotometer (Gilford Instrument Laboratories, Oberlin, Ohio). Hexosaminidase was purchased from Boehringer Mannheim Biochemicals, Indianapolis, Indiana, and p-nitrophenyl-l\r-acetyl-/3-D-glucosaminide from Sigma Chemical Company, St. Louis Missouri. The synthesis of the inhibitor, 2-acetamido-2,4-dideoxy-4-fluoro-D-galactono-1,5-lactone will be described in a paper under preparation.
RESULTS AND DISCUSSION
Both real data and contrived data were fit to the nine different models. The real data was from the inhibition of hexosaminidase by 2-acetamido-2,4dideoxy-Uluoro-D-galactono1,5-lactone, using p-nitrophenyl-N-acetyl-/3-Dglucosaminide as the substrate. The results from this experiment, comprising 46 data points, were fit to all four inhibition models. Sets of contrived data were fit to the other models. When fitting sets of data to any of the models without weighting, ROSFIT gave the same values to at least three significant figures for the parameters and the various statistics as did the commonly used mainframe statistical software packages BMDPAR, BMDP3R (ZO), and NONLIN (12). When sets of data were fit to the models with the iterative reweighting options, the values of the parameters usually agreed to only 1 or 2 significant figures. The complications of changing weighting factors at each iteration combined with somewhat different nonlinear regression algorithms and different convergence criteria in the various software packages probably led to these differences. The real inhibition data fit the unweighted competitive inhibition model best with K = 847 r 26.6 Z.L.Mand Ku = 27.2 r 0.876 PM. Figure 1 is a copy of the v vs A plot from the actual printout
NONLINEAR
REGRESSION
FOR MICROCOMPUTERS
!
43
113
0
1.86
2.12 CA3
3.18
4.25
. +
= .2
0
=0
=
5
FIG. 1. A copy of the actual printout from ROSFIT for the plot of velocity vs substrate concentration for the inhibition of hexosaminidase by 2-acetamido-2,Cdideoxy-Qfluoro-ogalactono- 1.5~lactone using p-nitrophenyl-N-acetyl-B-D-glucosaminide as the substrate. The units for velocity are change in absorbance per 30 min, and the units for substrate concentration A and inhibitor concentration Z are mM. The symbols used for the three concentrations of inhibitor are defined in the figure. The original printout has blue-dot matrix characters which do not photograph very well.
from this run. The quality of the printout is somewhat better than it appears in the figure; the blue dot matrix printing does not photograph well. The execution time of ROSFIT is adequate for enzyme kinetic applications. For example, excluding data entry, for the unweighted competitive inhibition example of Fig. 1, the total execution time was 18.5 min; 2 min for the 3 iterations, 14 min for the graph generation, and 2.5 min for miscellaneous operations. For a run with the Michaelis-Menten equation with 12 data points and the weighting factor of the reciprocal of the squared calculated velocity, the total execution time was about 4.5 min; 23 s for the 1 iteration, 2.5 min for the graph generation, and about 1.5 min for miscellaneous operations. The implementation of nonlinear regression algorithms on microcomputers and programmable calculators is relatively new, but expanding. Peck and Ban-et (19) adapted two previously written BASIC programs, one by Patlak and Pettigrew and one by Horowitz and Homer (20), to the Commodore PET microcomputer. Muri (21) has written a nonlinear regression program on a Texas Instrument TI-59 programmable calculator for fitting data to some simple pharmacokinetic models. Duggleby (22) has recently published a short general BASIC program for fitting data to two-parameter nonlinear models, which can be implemented on microcomputers. Koeppe and Hamann have also recently published a short general BASIC program for nonlinear curve fitting (23). We intend to update ROSFIT to include some more complicated enzyme kinetic models, and to include specific curve fitting programs for the analysis of data for ligand-macromolecule binding, radioimmunoassay, dose-response analysis, pharmacokinetics, transport studies, and survival analyses. ROSFIT and most future extensions of the package will consist of application-tailored programs. The major advantages of specific, application-tailored programs
44
GRECO ET AL.
over more general curve-fitting programs are that the user does not have to understand programming to use the program, the user does not have to do so much work with each run, and the graphical output can be optimized for each specific application. There were several other microcomputers considered for this project. They included the Apple II (Apple Computer, Cupertino, Calif.), Radio Shack TRS-80 (Tandy Corp., Fort Worth, Tex.), Texas Instrument 99/4 (Texas Instruments, Dallas, Tex.), and North Star Horizon (North Star Computers, Berkeley, Calif.). A system could have been configured around any of these micros instead of the HP-85 to complete this project. In fact, for the same price as that paid for the HP-85,” most of these systems would have included a minifloppy disk drive instead of a digital cassette tape recorder. But overall, the HP-85 was chosen because of its compactness, ease of programming, excellent documentation, good graphics, price, and miscellaneous features which make it suited for scientific purposes as opposed to business or hobby applications. Especially useful is the wide range of real numbers allowed with the HP-85. Real numbers are represented internally with 12 digits and a 3-digit exponent in the range -499 through 499. In summary, an enzyme kinetics curve-fitting package is now available for use on the HP-85 microcomputer. Advantages over conventional graphical analyses include increased accuracy and precision of parameter estimates, a savings of time, and an increased yield of useful information from experiments. Advantages over analyses using large mainframe computers include convenience, a possible savings of time if the mainframe is located outside of one’s laboratory, and graphics tailored specifically to a particular enzyme kinetic model. ROSFIT was written not just as an exercise to prove that sophisticated data analyses can be performed on a microcomputer norjust for data analyses in our laboratories. ROSFIT is intended for use by the biomedical community at large. Documentation on how to use the program and interpret the results is available. Instructions on how to receive a tape cassette containing ROSFIT and the documentation can be obtained by writing to the first author. ACKNOWLEDGMENTS We thank Dr. John Bertram and Dr. Jerry Roth for allowing the use of the microcomputers located in their laboratories. We thank Angela M. Di Loro for performing the hexosaminidase assays, and we thank Diana Fischer for running most of the analyses on the HP-85 and the Univac 90/60. This work was supported in part by Grants CA-16056 and CA-13038 from the U.S. Public Health Service. REFERENCES G. L., AND NIMMO, I. A. A comparison of seven methods for fitting the MichaelisMenten equation. Biochem. J. 149, 775 (1975).
1. ATKIN,
* The retail price of the HP-85 is $3250.
NONLINEAR
REGRESSION
FOR MICROCOMPUTERS
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I. A., AND ATKINS, G. L. Methods for fitting equations with two or more non-linear parameters. Biochem. J. 157, 489 (1976). GRECO, W. R., AND HAKALA, M. T. Evaluation of methods for estimating the dissociation constant of tight binding enzyme inhibitors. J. Biol. Chem. 254, 12104 (1979). LINEWEAVER, H., AND Buax, D. The determination of enzyme dissociation constants. J. Amer. Chem. Sot. 56, 658 (1934). WILKINSON, G. N. Statistical estimations in enzyme kinetics. Biochem. J. 80, 324 (1961). CLELAND, W. W. The Kinetics of enzyme-catalyzed reactions with two or more substrates or products. Biochim. Biophys. Acta 67, 173 (1963). CLELAND, W. W. The statistical analysis of enzyme kinetic data. In “Advances in Enzymology” (F. F. Nord, Ed.), Vol. 29, pp. l-32. Wiley, New York, 1967. CLELAND, W. W. Statistical analysis of enzyme kinetic data. In “Methods in Enzymology” (D. L. Purich, Ed.), Vol. 63, pp. 103-138. Academic Press, New York, 1979. ATKINS, G. L., AND NIMMO, I. A. Current trends in the estimation of Michaelis-Menten parameters. Biochem. 104, 1 (1980). RALSTON, M. In “BMDP Biomedical Computer Programs” (W. J. Dixon and M. B. Brown, Eds.), pp. 484-498. Univ. of Calif. Press, Berkeley, 1977. KNOTT, G. D. MLAB-A mathematical modeling tool. Camp. Prog. Biomed. 10, 271 (1979). METZLER, C. M., ELFRING, G. L., AND MC EWEN, A. J. A package of computer programs for pharmacokinetic modeling. Biometrics 30, 562 (1974). KOHN, M. C., MENTEN, L. E., AND GARFINKEL, D. A convenient computer program for fitting enzymatic rate laws to steady-state data. Camp. Biomed. Res. 12, 461 (1979). HANSON, K. R., LING, R., AND HAVIR, E. A computer program for fitting data to the Michaelis-Menten equation. Biochem. Biophys. Res. Comm. 29 (1967). SIANO, D. B., ZYSKIRD, J. W., AND FROMM, H. J. A computer program for fitting and statistically analyzing initial rate data applied to bovine hexokinase type II isozyme. Arch.
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16. MARQUARDT, D. W. An algorithm for least squares estimation of nonlinear parameters. J. Sot. Ind. Appl. Marh. 11, 431 (1963). 17. NASH, J. C. ‘Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation.” Wiley, New York, 1979. 18. VON FIGURA, K. Secretion of /3-Hexosaminidase by cultured human skin fibroblasts. Exp. Cell. Res. 111, 15 (1978). 19. PECK, C. C., AND BARRETT, B. B. Nonlinear least-squares regression program for microcomputers. .I. Pharmacokin. Biopharm. 7, 537 (1979). 20. HOROWITZ, D. L., AND HOMER, L. D. Analysis of biomedical data by time-sharing computers IV. Non-Linear regression analysis. Project No. MRO05: 200287, Rep. No. 25, Naval Med. Res. Inst. Nat. Naval Med. Ctr., Bethesda, Md. 20014, Feb. 26, 1970. 21. MUIR, K. T. Nonlinear least-squares regression analysis in pharmacokinetics: Application of a programmable calculator in model parameter estimation. Camp. Biomed. Res. 13, 307’(1980). 22. DUGGLEBY, R. G. A Nonlinear regression program for small computers. Anal. Biochem. 110,9 (1981). 23. KOEPPE, P., AND I&MANN, C. A program for non-linear regression analysis to be used on desk-top computers. Camp. Prog. Biomed. 12, 121 (1980).