- Email: [email protected]
Lucenz — programmes for undergraduate analysis of enzyme kinetic data
Biochemistry and Molecular Biology Education 28 (2000) 282}285
Lucenz * programmes for undergraduate analysis of enzyme kinetic data Alan G. Clark School of Biological Science, Victoria University of Wellington, P.O. Box 600, Wellington, New Zealand
Abstract Two programmes for analysis of enzyme kinetic data are described. The programmes have evolved from use in undergraduate classes and accordingly the emphasis is on simplicity in design and use, rather than on computational sophistication. Data "tting, to a selection of predetermined models, proceeds via a weighted linear regression on to the reciprocals of the substrate concentrations and catalytic rates and is rapid and robust. The programmes yield best-"t valus for K , V and, when appropriate, K together with K K G measures of the goodness of "t. Experimental data and lines of best "t may be presented graphically in up to seven dii!erent plotting methods. Entered data and the graphical and numerical results of model "tting are displayed on the same screen so that relationships between all three may be readily appreciated. Variants of the programmes may be employed in the Windows 3; or 9; platforms or in DOS. 2000 IUBMB. Published by Elsevier Science Ltd. All rights reserved.
LucenzII and LucenzIII are programmes for "tting enzyme kinetic data. They have been designed with the perceived needs of the average, modern biochemistry undergraduate in mind [1]. These programmes have been designed as a measure to reduce to a minimum the complexities of treating enzyme kinetic data, to allow students to concentrate on the essential information that can be extracted from such experiments. The "rst deliberate simpli"cation in these programmes is to "t the experimental data by weighted linear regression to double reciprocal data. This approach has been taken for the reasons outlined above and in full recognition of the superior theoretical merits of non-linear "tting to original data. To an increasing extent we have found that the average student "nds the theoretical basis of non-linear "tting di$cult to understand. In practice, such methods can be a source of frustration * particularly in view of the high frequency with which student analyses fail to converge, either due to poor data or to poor choice of initial estimates. Use of the linear transformation generates a process that is fast and robust: poor data generate clearly absurd results (to be discussed later), not an indeterminate state of non-convergence. The "tting proceeds via a single iteration of a matrix inversion by Gaussian elimination, as used by Cleland [2]. (This pro-
E-mail address: [email protected] (A.G. Clark).
cedure, rather than a simpler, purely algebraic approach [3], has been used since it permits the relatively straightforward conversion of the source code to yield a nonlinear "tting programme.)
1. Screen layout Simplicity in operation and in presentation has also been an aim. The programmes are written for Windows (3.1; or 9; * the optimal display is obtained with an 800;600 display) and all options are selected from pulldown menus. Once a procedure has started, all aspects of the programme are present on screen at the same time. (When "rst opened, some introductory instructions are displayed. These disappear as soon as "tting is started.) This feature permits students to see all necessary information, raw data and regression parameters, and thus to be able to identify dubious data quickly and easily. See Fig. 1. Reaction rate data are entered into a matrix. Substrate concentrations are entered into the column bordering the left-hand side of the matrix and co-substrate or inhibitor concentrations into the row bordering its upper edge. From there the data may be saved to "le. Data may also be retrieved from "le by conventional menu-driven procedures. Zero values for rates will be tolerated by the programmes but zero substrate concentration terms will
1470-8175/00/$20.00 2000 IUBMB. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 4 7 0 - 8 1 7 5 ( 0 0 ) 0 0 0 0 2 - 3
A.G. Clark / Biochemistry and Molecular Biology Education 28 (2000) 282}285
283
Fig. 1. The LuncenzIII screen display. All aspects of the programme are visible at the same time. The data matrix, graphical display and numerical results are shown in di!erent frames. The currently selected model may be changed by selecting di!erent options on the same screen. The data shown are simulated data generated by a model for competitive inhibition which incorporated a random 5% error into simulated substrate and enzyme concentrations.
not. Selection of mode (inhibited or uninhibited) is obtained by clicking on the appropriate menu item and selection of mechanism is made in a similar fashion. The "tting procedure is initiated by clicking on the `GOa button. The introductory instructions are replaced by frames in which the regression parameters and a graphical representation of the data points and the lines of best "t are shown. It needs to be stressed that all data are "tted at the same time: the lines of best "t correspond to a single-rate equation. A number of di!erent graphical representations of the data may be selected from a Plot menu. The screen lay-out allows the student to identify outliers and to judge almost instantaneously the e!ect of their removal on the values of regression parameters and the graphical representation. The students are encouraged to use the programme during the course of an experiment so as to identify dubious values that may need veri"cation. The weighting factor used is the square on the measured rate. A choice of weighting protocols has not been included as an option in order to maintain the greatest simplicity. Instructors may, however, wish to suggest that the students examine the e!ect of an un-weighted linear regression (e.g. as performed through proprietary spreadsheet or statistics packages) so that the e!ect of the weighting may be appreaciated.
2. Options The di!erent programmes, LucenzII and LucenzIII are designed for use at di!erent levels. LucenzII is used in an introductory course in protein and enzyme biochemistry. The number of choices on o!er is limited * the aim being to minimize confusion. Thus, only single-substrate kinetics are considered in the uninhibited case. The mechanisms of inhibition considered are competitive, non-competitive and uncompetitive inhibition. The inhibition is based on the model shown in Fig. 2 in which overall equilibrium is assumed. The corresponding equation is < K v" , 1#K /[S](1#[I]/K )#[I]/K Q GQ GG
(1)
Competitive inhibition is characterized by a "nte value for K with an in"nite value for K (the slope only is GQ GG a!ected). Uncompetitive inhibition has a "nite value for K whereas K has an in"nite value (vertical intercept GG GG a!ected by inhibitor concentration) and non-competitive inhibition is characterized by "nite values for both inhibition constants, both intercept and gradient being a!ected. In LucenzII, graphical transforms may be selected from v vs. [S]), the Lineweaver}Burke [4] double
284
A.G. Clark / Biochemistry and Molecular Biology Education 28 (2000) 282}285
Fig. 2. General model for inhibition used in the two Lucenz programmes. Overall equilibrium is assumed. K is the dissociation con? stant for the ES complex, K that for the EI complex and K that for GQ GG the EIS-to-ES dissociation.
reciprocal plot (1/v vs. 1/[S]), the Hanes [5] plot ([S]/v vs/ [S]), the Eadie}Hofstee plot (v/[S] vs. v) [6,7] and the Dixon [8] plot (1/v vs. [I]). The size of the data sets is limited to six points in LucenzII and, in the case of inhibition studies, up to three sets of data, obtained at di!ering inhibitor concentrations, may be used. Lucenz III has been designed for students in their third year who have already encountered LucenzII. It o!ers a greater range of models to "t to and greater range of graphical transformations. Three kinetic models may be selected in the case of uninhibited kinetics: single substrate, Ping}Pong Bi}Bi and sequential Bi}Bi. In the "rst case the regression parameters are V and K ; in the K K second, V , K , and K and in the third V , K , K and K ? @ K ? @ K where, in the case of an ordered mechanism, the G? subscript a refers to the "rst-bound substrate and b to the second and K is the dissociation constant for the G? enzyme}substrate A complex. In the case of inhibited kinetics, the same models are o!ered as in LucenzII. However, a greater range of plotting methods may be accessed, particularly for data from enzyme inhibition experiments. As well as those o!ered in LucenzII there are the methods of Cornish-Bowden [9] ([S]/v vs. [I]) and the Hunter and Downs plot [10] ([I]a/(1!a) vs. [S], where a"v /v, v being the rate in the presence of G the inhibitor and v the corresponding uninhibited rate)). The range of plotting methods available gives students a chance to try out the di!erent methods quickly easily and to judge for themselves the advantages and disadvantages of the di!erent methods. In keeping with the greater complexity of the models studied, a larger number of data points may be accommodated in LucenzIII: up to "ve sets of data may be entered, each containing up to 10 points.
These values are shown together with coe$cients of variaton. CVs have been chosen as an indication of goodness of "t with respect to each parameter, rather than the standard error, because they are normalized and no calculation on the part of the student is required to judge whether a parameter is well determined (CV(0.2) or not. In addition two parameters are shown indicating the overall goodness of "t. The "rst is the square root of the weighted mean of the squared residuals, in which the weighting factor is v [3]. The second is closely related: G the root-mean-square fractional residual. This gives a measure of the size of the average discrepancy between experimental and theoretical values, measured as a fraction of the theoretical value. Values of less than about 0.15 indicate a good to reasonable "t. Values greater than this indicate a relatively poor "t. These parameters are used in judging the relative goodness of "t to di!ering mechanistic models. Thus, large values for coe$cients of variation and the RMS fractional residual (say '0.2), indicate that a di!erent model needs to be tried, as do absurd results such as negative values for limiting velocities or dissociation constants or values for the latter much larger than the range of substrate concentrations employed. The last result might, of course, indicate that an inappropriate range of substrate concentrations had been used.
4. Recording results Numerical results displayed in the kinetic parameters frame may be selected and copied to the clipboard from which they may be incorporated into a word processing document. The data in the graphics display frame may also be copied to the clipboard and pasted into hard copy. The graphics display frame may be resized, up to full screen, and clicking the plot button re-sizes the display to "ll the frame.
5. Help 5les For both LucenzII and LucenzIII there are extensive help "les providing information about the theory underlying enzyme kinetics and also about how to accomplish the di!erent tasks that can be performed through the programme.
6. Experiments 3. Presentation of results In both programmes, after the "tting procedure, the best-"t values of the regression parameters are shown in the frame in the top, right-hand corner of the screen.
These programmes may be used in connection with any simple enzyme kinetic experiment suitable for undergraduate laboratory classes. For instance, LucenzIII is used in a third year biochemistry laboratory class to analyse results obtained from an examination of the
A.G. Clark / Biochemistry and Molecular Biology Education 28 (2000) 282}285
reaction catalysed by rabbit muscle M4 lactate dehydrogenase in the presence of varying concentrations of lactic acid and of NAD>. The e!ect of oxalate as an inhibitor competing with lactate is examined. By characterising the e!ects of the inhibitor when either substrate is varied it is possible (at least in theory), to deduce the order of binding of the substrates in the course of the reaction [11,12].
285
some, if the colour depth is too low, the graphics display will show all the theoretical lines, but only a limited number of the experimental points. The reason for this peculiar behaviour is not known but can generally be recti"ed by setting the display to the highest possible number of colours.
Acknowledgements 7. Software The installation software for LucenzII and LucenzIII may be downloaded from the site http://www.vuw.ac.nz/ &clarkag/teaching as self-extracting .exe "les. They should be downloaded into a convenient temporary directory and unzipped by clicking on the exe "le. Then clicking on setup.exe initiates the "nal installation. The desired "nal location for the programme will be asked for during the installation process. Versions of these programmes are available for both Windows 3; and 9; platforms. All come with associated Help "les. Also included are sets of data simulating all the models that these programmes can "t data to. This permits the ready testing of the software for proper function and also allows students to familiarize themselves with the programmes and to examine the di!erent plotting methods. Also available at this site are equivalent packages (tchin2.zip and tchin3.zip) running on the DOS platform.
8. Problems The only major problem that has been encountered so far appears to derive from certain colour cards. With
During the development of these programmes over a period of some years, the author has received much helpful feedback from colleagues who have used these programmes in teaching. In particular, that from Dr. T.W. Jordan and Dr. G.K. Chambers is gratefully acknowledged.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
A. McDonald, K. Tipton, The Biochemist 21 (1999) 18}23. W.W. Cleland, Adv. Enzymol. 29 (1969) 1}32. G.N. Wilkinson, Biochem. J. 80 (1961) 324}332. H. Lineweaver, D. Burk, J. Am. Chem. Soc. 56 (1934) 658}666. C.S. Hanes, Biochem. J. 26 (1932) 1406}1421. G.S. Eadie, J. Biol. Chem. 146 (1942) 85}93. B.H.J. Hofstee, J. Biol. Chem. 199 (1952) 357}364. M. Dixon, Biochem. J. 55 (1953) 170}171. A. Cornish-Bowden, Biochem. J. 137 (1974) 143}144. A. Hunter, C. Downs, J. Biol. Chem. 157 (1945) 427}446. V. Zewe, H.J. Fromm, J. Biol. Chem. 237 (1962) 1668}1675. V. Zewe, H.J. Fromm, Biochemistry 4 (1965) 782}792.
Lucenz * programmes for undergraduate analysis of enzyme kinetic data Alan G. Clark School of Biological Science, Victoria University of Wellington, P.O. Box 600, Wellington, New Zealand
Abstract Two programmes for analysis of enzyme kinetic data are described. The programmes have evolved from use in undergraduate classes and accordingly the emphasis is on simplicity in design and use, rather than on computational sophistication. Data "tting, to a selection of predetermined models, proceeds via a weighted linear regression on to the reciprocals of the substrate concentrations and catalytic rates and is rapid and robust. The programmes yield best-"t valus for K , V and, when appropriate, K together with K K G measures of the goodness of "t. Experimental data and lines of best "t may be presented graphically in up to seven dii!erent plotting methods. Entered data and the graphical and numerical results of model "tting are displayed on the same screen so that relationships between all three may be readily appreciated. Variants of the programmes may be employed in the Windows 3; or 9; platforms or in DOS. 2000 IUBMB. Published by Elsevier Science Ltd. All rights reserved.
LucenzII and LucenzIII are programmes for "tting enzyme kinetic data. They have been designed with the perceived needs of the average, modern biochemistry undergraduate in mind [1]. These programmes have been designed as a measure to reduce to a minimum the complexities of treating enzyme kinetic data, to allow students to concentrate on the essential information that can be extracted from such experiments. The "rst deliberate simpli"cation in these programmes is to "t the experimental data by weighted linear regression to double reciprocal data. This approach has been taken for the reasons outlined above and in full recognition of the superior theoretical merits of non-linear "tting to original data. To an increasing extent we have found that the average student "nds the theoretical basis of non-linear "tting di$cult to understand. In practice, such methods can be a source of frustration * particularly in view of the high frequency with which student analyses fail to converge, either due to poor data or to poor choice of initial estimates. Use of the linear transformation generates a process that is fast and robust: poor data generate clearly absurd results (to be discussed later), not an indeterminate state of non-convergence. The "tting proceeds via a single iteration of a matrix inversion by Gaussian elimination, as used by Cleland [2]. (This pro-
E-mail address: [email protected] (A.G. Clark).
cedure, rather than a simpler, purely algebraic approach [3], has been used since it permits the relatively straightforward conversion of the source code to yield a nonlinear "tting programme.)
1. Screen layout Simplicity in operation and in presentation has also been an aim. The programmes are written for Windows (3.1; or 9; * the optimal display is obtained with an 800;600 display) and all options are selected from pulldown menus. Once a procedure has started, all aspects of the programme are present on screen at the same time. (When "rst opened, some introductory instructions are displayed. These disappear as soon as "tting is started.) This feature permits students to see all necessary information, raw data and regression parameters, and thus to be able to identify dubious data quickly and easily. See Fig. 1. Reaction rate data are entered into a matrix. Substrate concentrations are entered into the column bordering the left-hand side of the matrix and co-substrate or inhibitor concentrations into the row bordering its upper edge. From there the data may be saved to "le. Data may also be retrieved from "le by conventional menu-driven procedures. Zero values for rates will be tolerated by the programmes but zero substrate concentration terms will
1470-8175/00/$20.00 2000 IUBMB. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 4 7 0 - 8 1 7 5 ( 0 0 ) 0 0 0 0 2 - 3
A.G. Clark / Biochemistry and Molecular Biology Education 28 (2000) 282}285
283
Fig. 1. The LuncenzIII screen display. All aspects of the programme are visible at the same time. The data matrix, graphical display and numerical results are shown in di!erent frames. The currently selected model may be changed by selecting di!erent options on the same screen. The data shown are simulated data generated by a model for competitive inhibition which incorporated a random 5% error into simulated substrate and enzyme concentrations.
not. Selection of mode (inhibited or uninhibited) is obtained by clicking on the appropriate menu item and selection of mechanism is made in a similar fashion. The "tting procedure is initiated by clicking on the `GOa button. The introductory instructions are replaced by frames in which the regression parameters and a graphical representation of the data points and the lines of best "t are shown. It needs to be stressed that all data are "tted at the same time: the lines of best "t correspond to a single-rate equation. A number of di!erent graphical representations of the data may be selected from a Plot menu. The screen lay-out allows the student to identify outliers and to judge almost instantaneously the e!ect of their removal on the values of regression parameters and the graphical representation. The students are encouraged to use the programme during the course of an experiment so as to identify dubious values that may need veri"cation. The weighting factor used is the square on the measured rate. A choice of weighting protocols has not been included as an option in order to maintain the greatest simplicity. Instructors may, however, wish to suggest that the students examine the e!ect of an un-weighted linear regression (e.g. as performed through proprietary spreadsheet or statistics packages) so that the e!ect of the weighting may be appreaciated.
2. Options The di!erent programmes, LucenzII and LucenzIII are designed for use at di!erent levels. LucenzII is used in an introductory course in protein and enzyme biochemistry. The number of choices on o!er is limited * the aim being to minimize confusion. Thus, only single-substrate kinetics are considered in the uninhibited case. The mechanisms of inhibition considered are competitive, non-competitive and uncompetitive inhibition. The inhibition is based on the model shown in Fig. 2 in which overall equilibrium is assumed. The corresponding equation is < K v" , 1#K /[S](1#[I]/K )#[I]/K Q GQ GG
(1)
Competitive inhibition is characterized by a "nte value for K with an in"nite value for K (the slope only is GQ GG a!ected). Uncompetitive inhibition has a "nite value for K whereas K has an in"nite value (vertical intercept GG GG a!ected by inhibitor concentration) and non-competitive inhibition is characterized by "nite values for both inhibition constants, both intercept and gradient being a!ected. In LucenzII, graphical transforms may be selected from v vs. [S]), the Lineweaver}Burke [4] double
284
A.G. Clark / Biochemistry and Molecular Biology Education 28 (2000) 282}285
Fig. 2. General model for inhibition used in the two Lucenz programmes. Overall equilibrium is assumed. K is the dissociation con? stant for the ES complex, K that for the EI complex and K that for GQ GG the EIS-to-ES dissociation.
reciprocal plot (1/v vs. 1/[S]), the Hanes [5] plot ([S]/v vs/ [S]), the Eadie}Hofstee plot (v/[S] vs. v) [6,7] and the Dixon [8] plot (1/v vs. [I]). The size of the data sets is limited to six points in LucenzII and, in the case of inhibition studies, up to three sets of data, obtained at di!ering inhibitor concentrations, may be used. Lucenz III has been designed for students in their third year who have already encountered LucenzII. It o!ers a greater range of models to "t to and greater range of graphical transformations. Three kinetic models may be selected in the case of uninhibited kinetics: single substrate, Ping}Pong Bi}Bi and sequential Bi}Bi. In the "rst case the regression parameters are V and K ; in the K K second, V , K , and K and in the third V , K , K and K ? @ K ? @ K where, in the case of an ordered mechanism, the G? subscript a refers to the "rst-bound substrate and b to the second and K is the dissociation constant for the G? enzyme}substrate A complex. In the case of inhibited kinetics, the same models are o!ered as in LucenzII. However, a greater range of plotting methods may be accessed, particularly for data from enzyme inhibition experiments. As well as those o!ered in LucenzII there are the methods of Cornish-Bowden [9] ([S]/v vs. [I]) and the Hunter and Downs plot [10] ([I]a/(1!a) vs. [S], where a"v /v, v being the rate in the presence of G the inhibitor and v the corresponding uninhibited rate)). The range of plotting methods available gives students a chance to try out the di!erent methods quickly easily and to judge for themselves the advantages and disadvantages of the di!erent methods. In keeping with the greater complexity of the models studied, a larger number of data points may be accommodated in LucenzIII: up to "ve sets of data may be entered, each containing up to 10 points.
These values are shown together with coe$cients of variaton. CVs have been chosen as an indication of goodness of "t with respect to each parameter, rather than the standard error, because they are normalized and no calculation on the part of the student is required to judge whether a parameter is well determined (CV(0.2) or not. In addition two parameters are shown indicating the overall goodness of "t. The "rst is the square root of the weighted mean of the squared residuals, in which the weighting factor is v [3]. The second is closely related: G the root-mean-square fractional residual. This gives a measure of the size of the average discrepancy between experimental and theoretical values, measured as a fraction of the theoretical value. Values of less than about 0.15 indicate a good to reasonable "t. Values greater than this indicate a relatively poor "t. These parameters are used in judging the relative goodness of "t to di!ering mechanistic models. Thus, large values for coe$cients of variation and the RMS fractional residual (say '0.2), indicate that a di!erent model needs to be tried, as do absurd results such as negative values for limiting velocities or dissociation constants or values for the latter much larger than the range of substrate concentrations employed. The last result might, of course, indicate that an inappropriate range of substrate concentrations had been used.
4. Recording results Numerical results displayed in the kinetic parameters frame may be selected and copied to the clipboard from which they may be incorporated into a word processing document. The data in the graphics display frame may also be copied to the clipboard and pasted into hard copy. The graphics display frame may be resized, up to full screen, and clicking the plot button re-sizes the display to "ll the frame.
5. Help 5les For both LucenzII and LucenzIII there are extensive help "les providing information about the theory underlying enzyme kinetics and also about how to accomplish the di!erent tasks that can be performed through the programme.
6. Experiments 3. Presentation of results In both programmes, after the "tting procedure, the best-"t values of the regression parameters are shown in the frame in the top, right-hand corner of the screen.
These programmes may be used in connection with any simple enzyme kinetic experiment suitable for undergraduate laboratory classes. For instance, LucenzIII is used in a third year biochemistry laboratory class to analyse results obtained from an examination of the
A.G. Clark / Biochemistry and Molecular Biology Education 28 (2000) 282}285
reaction catalysed by rabbit muscle M4 lactate dehydrogenase in the presence of varying concentrations of lactic acid and of NAD>. The e!ect of oxalate as an inhibitor competing with lactate is examined. By characterising the e!ects of the inhibitor when either substrate is varied it is possible (at least in theory), to deduce the order of binding of the substrates in the course of the reaction [11,12].
285
some, if the colour depth is too low, the graphics display will show all the theoretical lines, but only a limited number of the experimental points. The reason for this peculiar behaviour is not known but can generally be recti"ed by setting the display to the highest possible number of colours.
Acknowledgements 7. Software The installation software for LucenzII and LucenzIII may be downloaded from the site http://www.vuw.ac.nz/ &clarkag/teaching as self-extracting .exe "les. They should be downloaded into a convenient temporary directory and unzipped by clicking on the exe "le. Then clicking on setup.exe initiates the "nal installation. The desired "nal location for the programme will be asked for during the installation process. Versions of these programmes are available for both Windows 3; and 9; platforms. All come with associated Help "les. Also included are sets of data simulating all the models that these programmes can "t data to. This permits the ready testing of the software for proper function and also allows students to familiarize themselves with the programmes and to examine the di!erent plotting methods. Also available at this site are equivalent packages (tchin2.zip and tchin3.zip) running on the DOS platform.
8. Problems The only major problem that has been encountered so far appears to derive from certain colour cards. With
During the development of these programmes over a period of some years, the author has received much helpful feedback from colleagues who have used these programmes in teaching. In particular, that from Dr. T.W. Jordan and Dr. G.K. Chambers is gratefully acknowledged.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
A. McDonald, K. Tipton, The Biochemist 21 (1999) 18}23. W.W. Cleland, Adv. Enzymol. 29 (1969) 1}32. G.N. Wilkinson, Biochem. J. 80 (1961) 324}332. H. Lineweaver, D. Burk, J. Am. Chem. Soc. 56 (1934) 658}666. C.S. Hanes, Biochem. J. 26 (1932) 1406}1421. G.S. Eadie, J. Biol. Chem. 146 (1942) 85}93. B.H.J. Hofstee, J. Biol. Chem. 199 (1952) 357}364. M. Dixon, Biochem. J. 55 (1953) 170}171. A. Cornish-Bowden, Biochem. J. 137 (1974) 143}144. A. Hunter, C. Downs, J. Biol. Chem. 157 (1945) 427}446. V. Zewe, H.J. Fromm, J. Biol. Chem. 237 (1962) 1668}1675. V. Zewe, H.J. Fromm, Biochemistry 4 (1965) 782}792.