Modelling Metabolism with Mathematica PDF

Preface

Simulation of Metabolism

The experimental and theoretical study of metabolism in mammalian cells, and the human erythrocyte in particular, has a long history, so it is valid to challenge the need for another book on this topic; but as you will see, this book is very different from previous ones in that it is interactive. Our response is that an understanding of cellular metabolism at the molecular level, with all its intricate controls, is far from complete and many fundamental and clinically relevant discoveries remain to be made. Three major technological advances in recent years – mass spectrometry, NMR spectroscopy, and computing – have greatly contributed to the renaissance of cellular metabolism as a topic of research. It is the computer modelling of metabolism, in particular the simulation of time courses of reactions, that is our focus.

Anticipated Readership

This book is aimed at advanced undergraduate and postgraduate students of biochemistry, enzymology, functional genomics, biotechnology, theoretical biology, computational science, applied mathematics – indeed, anyone interested in applying computational methods to the simulation and study of metabolic systems.

Contents

Chapters 1 and 2 provide an introduction to biochemical enzyme kinetics, including basic definitions; the mathematical formulation of reaction schemes and the computerbased methods used in their analysis; and quantitative aspects of enzymology such as the analysis of kinetic data, the mechanistic basis of enzyme inhibitions, and models of enzyme regulation. Chapter 3 contains an introduction to the procedures used to simulate metabolic systems using symbolic computation; and a model of the urea cycle of the human liver is used to exemplify these. More advanced methods incorporating matrix algebra are introduced in Chapter 4, where we show that the differential rate equations describing complex metabolic reaction schemes can be represented in a simple and compact way. Our implementation of the key elements of metabolic control analysis (MCA) as presented by Heinrich and SchusterH1L is described in Chapter 5. Parameter estimation is the subject of Chapter 6 and here we consider linear and nonlinear least-squares regression analysis for this purpose, and parameter estimation in large-scale metabolic networks where over-parameterization may be an issue. Chapter 7 applies the theory and methods developed in the previous chapters to the human erythrocyte to illustrate how a realistic model of metabolism can be built up. Finally, the concepts and methods described in Chapters 5 and 6 are used to perform MCA on the erythrocyte model presented in Chapter 7. The five appendices contain supplementary material and Mathematica code that is required to run the programs and worked examples contained in Chapters 1 – 8 from the interactive CD.

Layout

Each chapter is divided into sections for easy cross-referencing between topics. We have made extensive use of worked examples to emphasize, or to extend, a basic concept that is discussed in the body of the text. The exercises are posed as questions and are made to stand out from the main text by the use of a solid horizontal line and the letters Q (question) and A (answer) in the margin. This approach has been adapted from that used in our ownH2L and other successful Schaum's Outline texts. In many cases the references that we cite are merely representative of the literature in a particular topic; we apologize in advance to those of our colleagues who feel their work has not been adequately referenced.

Motivation

Our motivation for writing this book was our success in modelling a particularly troublesome aspect of human erythrocyte metabolism, and our wish to share the methodology that led to the insights that we consider would have been unattainable by other means. Briefly, these insights were as follows.(3-5) Sound explanations for several readily elicited metabolic responses in human erythrocytes had not previously been found, notably, the exquisite sensitivity of the steady-state concentration of 2,3-bisphosphoglycerate (2,3BPG) to pH, and to changes in oxygen partial pressure. Alteration of the intracellular pH from the normal value of 7.2 to 6.8 (only 0.4 pH units), which is a transition that is frequently encountered physiologically, brings about, over several hours, an almost total disappearance of 2,3BPG. Researchers sought an unidentified effector molecule that would be produced in a reaction that is under the control of the state of oxygenation of hemoglobin. It was surmised that this compound might decrease the activity of 2,3BPG synthase, or activate the 2,3BPG phosphatase that catalyses its hydrolysis, thus linking the oxygen partial pressure to the 2,3BPG concentration. On the other hand, we posited that an explanation of these two notable metabolic responses might be found by simply piecing together the vast and disparate metabolic and kinetic data available from almost a century of relevant scientific literature, and from our own experiments using modern analytical techniques. Hence, we combined our own NMR spectroscopy data of whole cells and cell extracts with computer modelling of the metabolism and provided a plausible explanation for the observations.

NMR experiments of many types provide a means of rapidly, precisely, and in some circumstances uniquely, obtaining estimates of metabolite concentrations in a totally non-invasive way. As such, NMR methods admirably satisfy Krebs’ notion of what constitutes the essential ingredient for scientific progress. Specifically, in summarizing his discovery of the urea cycle, Krebs wrote :

"If there is a lesson to be drawn … it is … the importance to progress of new techniques, especially techniques which make it possible to conduct a large number of experiments, and of studying a phenomenon under many different conditions. … It also illustrates the importance of following up an unexpected and puzzling observation arising in the course of the experiment. Luck, it is true, is necessary, but the more experiments are carried out, the greater is the probability of meeting with luck.

Another "method" that allows rapid evaluation of experimental data should be added to the list of techniques that Krebs did not use, namely, computer-based simulation of metabolic systems. We are not, of course, the first to suggest this approach. It was pioneered at the University of Philadelphia in the 1950s by Britton Chance and Joseph Higgins, and then in a very elaborate way by David and Lillian Garfinkel.H7L In the 1970s one of us (PWK) helped construct a computer model of the human urea cycle. However, a persistent criticism of this and all other computer models of metabolism has been the failure to address the consequences of intracellular partitioning of metabolites between the cytoplasm, mitochondria, and other organelles, and likely effects of the viscous intracellular milieu and surface adhesion, on the rates of enzymic reactions. Because it is non-invasive and highly selective to the detection of chemical species, NMR spectroscopy has provided a means to address many of these questions.

Also, in the present decade computers have gained so much in calculating speed and user friendliness that it is now routine practice, using a modest-cost personal computer, to calculate the time dependence of a kinetic system, whether physical, chemical, or biochemical, that is described by arrays of hundreds of stiff non-linear differential equations. In the past, the solutions were obtained using specialized programs, often written in machine code by the individual scientist. However, with the advent of sophisticated general programming environments like Mathematica that have implementations of contemporary algorithms and excellent graphics output capabilities, the task of developing new models of metabolism and visualizing their responses has become accessible to many students of biochemistry, and the life sciences in general.