Analysis of Enzyme Kinetic Data

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1 Analysis of Enzyme Kinetic Data

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3 Analysis of Enzyme Kinetic Data ATHEL CORNISH-BOWDEN Directeur de Recherche Émérite, Centre National de la Recherche Scientifique, Marseilles OXFORD UNIVERSITY PRESS 1995 This file was prepared for distribution after the printed edition of the book went out of print in It was prepared from the original files sent by the author to the publisher, not from the files after editing the the publisher s office and actually used for printing the book. There may therefore be minor variations in wording from the published version, and, more important, there may be some errors that were in the original files but which were corrected during printing of the book. In any case I should be grateful to have errors brought to my attention. At present my address is However, this changes from time to time so you may need to consult my web site at to determine how to contact me. The pagination in this file follows that in the book, so p. 100, for example, contains the same material as appears on p. 100 of the book. That is why most pages end in the middle of a line. This file is Athel Cornish-Bowden However, permission is hereby granted to make use of it for personal study, research, teaching, etc., without restriction. It will be appreciated, however, if any use of it in published work is apopropriately acknowledged therein by referring to the original book. As the computer program Leonora included with the book was written for an operating system that is now obsolete, the program itself has become difficult to use, and unappealing to people who are more familiar with more modern operating systems. Chapters 8 11 of the book are therefore omitted from this file. At the moment the figures are scanned from the book (as the original files used to create them proved to be unusable). In due course, and if there is sufficient interest, I shall replace them with better versions, and I shall also try to root out and correct errors in the text. Whether this happens or not will depend on whether I get any feedback. If no one writes to tell me that they have found this file useful (or otherwise) then it will remain for ever in the form you see now. Version 1.2, 16 November 2004 (minor corrections to this page, 19 October 2009, 28 August 2014)

4 Preface This is a book that I have wanted to write (the first six chapters, at least) for many years, and, indeed, I made a start on an ancestral version during a sabbatical in However, it soon became clear that a short book on the theory of data analysis in enzymology would have very limited appeal, and for this and various other reasons the original project did not advance very far. The arrival of the personal computer has completely transformed the world of scientific computing, however, to the point where virtually every working scientist is now also a computer user. As a result, it has become quite feasible to incorporate all of the methods of analysis developed in the 1960s and 1970s into a single program and to present both the methods and the program in a single book. The two principal parts of the book are largely independent of one another, with only a short link section (Chapter 7) between them: the first six chapters provide a theoretical account of statistical analysis of kinetic data for enzyme-catalysed reactions in the steady state; the last four describe Leonora, a program for analysing enzyme kinetic data on the IBM PC and compatible computers. Each of these parts can be read almost independently of the other, and each makes very little reference to the other. One may reasonably ask, therefore, why they have been bound between the same pair of covers and offered as a single book. The answer is that although they can be read in isolation from one another, that is far from being the best way to proceed. Something that will strike anyone who pays more than passing attention to statistics journals is that the number of statistical methods that have been proposed for scientists and engineers to use is much larger than the number of such methods that are actually in use by scientists or engineers. This is less true of methods proposed in journals that are normally read by their potential users, but it is still true to some degree of methods of kinetic data analysis that one can find in the biochemical literature. It is one thing to be reasonably convinced by a research article that a new method is better than existing ones, but it is quite another to go out and use it in the laboratory if one has to develop it from nothing. The reason for adding the practical part of this book (and the accompanying software) to the theoretical part, therefore, is to provide the tools necessary for the reader to test and apply all of the theory. This leaves unanswered, however, the complementary question of why the user of the software would want to be bothered with the theory. The

5 reason is in the sort of program that Leonora is. It does not follow the philosophy of assuming that there is One True Way of analysing data that must be applied in all circumstances. On the contrary, it offers a great deal of choice to its users, though to avoid making use too difficult it makes its own (i.e. my) choices when others are not made. To make appropriate choices the user needs a theoretical point of reference. Moreover, when I started writing Leonora I intended it to permit use of virtually any method the user might wish to try, but in practice the number of possibilities is almost infinite, because the more choices allowed, the more sub-choices these imply, etc. Consequently Leonora does make some restrictions, but to know why these restrictions apply rather than others one again needs a point of reference. Not all of the methods Leonora offers are in my opinion good methods, and even if they were it would be reasonable to ask what is the point of offering so many. This comes back to the question of user choice: far too many programs of all kinds are written in the spirit of the One True Way, and when they tolerate different preferences from those of the programmer they may force them to be specified every time the program is used. In the case of enzyme kinetics some of the most widely used methods come into the category of bad methods, but the solution is not to forbid their use if potential users don t find the methods they find most natural they won t continue to be users but to try to persuade users that better methods exist that are just as convenient. The ideal, in my view, is not only to offer a choice, but also to offer users who don t want to avail themselves of the choice a default method that will work well in most circumstances. Anyone using Leonora to analyse results of research experiments will, in all likelihood, settle quite soon on one method of analysis and ignore the others. Not everyone is a resarcher, however, and for teaching the principles of data analysis there is a more obvious need for a program that will allow the use and comparison of many different methods. Leonora is intended to address this need. I am grateful to Faculty of Sciences of the University of Chile for appointing me on two occasions to the visiting Chair set up in memory of the late Professor Hermann Niemeyer Fernández, and to the members of the Laboratory of Biochemistry in Santiago for providing me with the opportunity to do much of the work on this book there. Work on some of the methods described in the book benefitted greatly from collaborations with Robert Eisenthal and Laszlo Endrenyi, and I thank both of them for this. Finally, I thank Véronique Raphel for allowing me to use data from her doctoral thesis as the practical example around which Chapter 7 is written.

6 Contents I. THEORY 1. Least-Squares Analysis: Basic Principles 1.1 The Statistical Approach to Data Analysis 1.2 The Continuing Importance of Graphs 1.3 True Values, Population Values, Observed Values and Estimates 1.4 Variance 1.5 Weighting 1.6 Fitting the Straight Line 1.7 Degrees of Freedom 1.8 Choice of Dependent Variable 2. Fitting the Michaelis Menten Equation by Least Squares 2.1 Linearization of the Michaelis Menten Equation 2.2 Corresponding Results for the Double-Reciprocal Plot 2.3 Choosing the Proper Weights for the Rate 2.4 Standard Errors of Michaelis Menten Parameters 3. More than Two Parameters 3.1 The General Linear Model 3.2 Standard Errors in the General Linear Model 3.3 Application to Enzyme Inhibition and Other Kinetic Examples 3.4 Comparing Models 3.5 Additional Remarks about Residual Plots 3.6 Use of Replicate Observations 4. Maximum Likelihood and Efficiency 4.1 The Theoretical Basis of Least Squares 4.2 Minimum Variance 4.3 The Normal Distribution 4.4 How Normal is the Normal Distribution? 4.5 Efficiency 4.6 The Central Limit Theorem 4.7 Review of Assumptions Implicit in Least Squares

7 5. Generalized Medians: Looking beyond Least Squares 5.1 Doing without Information on Distributions and Weights 5.2 Median Estimate of the Slope of a Straight Line 5.3 Confidence Limits for Median Slope Estimates 5.4 Relationship between Least-squares and Median Estimates 5.5 Median Estimates of Michaelis Menten Parameters 5.6. Least Absolutes Fitting 6. Robust Regression 6.1 Recognizing and Dealing with Outliers 6.2 Biweight Regression 6.3 Assessing Heteroscedasticity 6.4 Worked Example of Robust Regression 6.5 The Jackknife and Bootstrap 6.6 Minimax Fitting 6.7 Reading the Statistics Literature 7. Analysis of an Example II. INTERLUDE 7.1 Introduction: Acylaminoacyl-peptidase 7.2 Preliminary Examination of the Data 7.3 Inhibition by Acetyl-L-alanine 7.4 Inhibition by Acetyl-D-alanine 7.5 Planning Future Experiments III. PRACTICE [not included in this file] 8. Leonora: a Program for Robust Regression of Enzyme Data 8.1 Introduction 8.2 Typographical Conventions 8.3 Installation on a Hard Disk 8.4 Example 1: the Michaelis Menten Equation 8.5 Example 2: Competitive Inhibition 8.6 Two-substrate Kinetics 8.7 Example 3: ph-dependence Data 8.8 Fitting Other Equations 8.9 Screen Layout 8.10 Miscellaneous Points

8 9. Leonora Menus 9.1 Main Menu 9.2 Data Menu Editing Menu Weighting Function Window 9.3 Equation Menu Common Entries in the Equation Menu Specific Entries in the Equations Menu Adding a New Entry to the Equations Menu 9.4 Output Requirements Menu 9.5 Calculations Menu Methods Menu Weighting System Menu 9.6 Plotting Menu 9.7 Graphical Menu 9.8 Setting Defaults 10. Customizing Leonora 10.1 Introduction and Warning 10.2 Editing a Menu 10.3 Editing a Warning 10.4 Editing Other Messages 10.5 Editing Help Files 10.6 Editing the Equation File 11. Use of Simulated Data 11.1 Introduction: Generation of Pseudo-Random Numbers 11.2 Changing the Distribution of Pseudo-Random Numbers 11.3 Simulating Leonora 11.4 Entering Data 11.5 Selecting Equations 11.6 True Parameter Values, Error Parameters, Output 11.7 Methods and Weights 11.8 Results Screen One Equation, one Method Two Equations More than two Equations More than one Method Replicate Observations 11.9 Randomizer References Index

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