Modeling Metabolism with Mathematica_ Detailed Examples Including Erythrocyte Metabolism [Mulquiney & Kuchel 2003-05-14].pdf

Library of Congress Cataloging-in-Publication Data

Mulquiney, Peter J.

Modeling metabolism with Mathematica / Peter J. Mulquiney, Philip W. Kuchel.

p. cm.

Includes bibliographical references and index.

ISBN 0-8493-1468-2 (alk. paper)

1. Cell metabolismÐComputer simulation. 2. Enzyme kineticsÐComputer simulation.

3. ErythrocytesÐComputer simulation. 4. Mathematica (Computer program language)

I. Kuchel, Philip W. II. Title.

QH634.5.M85 2003

572

0113--dc21

2003046279

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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 Î mass spectrometry, NMR spectroscopy,

and computing Î 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 Î 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 computer-

based 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 Schuster

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 Î 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 own

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 KrebsÓ 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 È it is È 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. È 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.

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.

Acknowledgments

This book has emerged from work carried out by past and present students in PWKÓs

laboratory: accordingly, we gratefully acknowledge Michael York, Zoltan Endre, Glenn

King, David Thorburn, Kiaran Kirk, Julia Raftos, Lisa McIntyre, Nicola Nygh, Serena

Hyslop, Lindy Rae, and Hilary Berthon. They contributed kinetic data primarily from

NMR spectroscopy and also assisted in the development of the computer model of

human erythrocyte metabolism that culminated in PJM's Ph.D. thesis. We also thank

Bill Bubb and Bob Chapman for their invaluable input into the NMR work, and Brian

Bulliman and Bill Lowe for computing and technical assistance, respectively. David

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