Drs. Anna and T T Wu



T T's new book, published by Kluwer Academic
ISBN 0-7923-7447-9

He dedicated it to "all my students of the last forty years"
 
 
 

PREFACE

This book is designed to illustrate the importance of analytical methods applied to some basic molecular biology problems. It serves two purposes: (1) for molecular biologists to learn some mathematics, and (2) for applied mathematicians to learn some molecular biology. As our starting point, some knowledge of both mathematics and molecular biology has to be assumed. We would like to begin at the level of high school advanced placement courses. In addition, it is important to try to solve fundamental molecular biology problems with simple applied mathematical methods, rather than looking for complicated mathematical equations in less important problems in biology.

To illustrate this approach, eight fundamental problems in molecular biology will be discussed in this book in the order of mathematical complexity. There are, of course, many more other important problems.  However, after going over this book, the readers can develop their own analytical methods to study any problem of particular interest to them. The basic philosophy is that we are going to start with crucial experimental data for a specific problem in molecular biology. A simple model will then be constructed with the inclusion of as much biological facts as possible. From this model, explicit predictions can then be deduced and in turn suggest further experimental studies.

Chapter I discusses the important experiments on amino acid sequencing of Bence Jones proteins, their relationship to antibodies or immunoglobulins, alignment of these sequences, calculation of variability as function of position to locate antibody combing sites, structures of antibody-antigen complexes, humanization of rodent antibodies for therapeutic use, etc.

Chapter 2 considers the saturation of hemoglobin with oxygen at equilibrium. This classical problem has been analyzed by many famous molecular biologists. A particular set of extremely accurate experimental measurements will be graphed in the Hill’s plot. Three models, one-constant (Hill), two-constant (Paulin) and three-constant (MWC), are discussed in detail. Non-linear least square fitting of these theoretical equations to the experimental data in the Hill’s plot gives the equilibrium constants indicative of allosteric transition. Further experiments provide a detailed molecular mechanism of this transition.  In order to sequence the entire genome of Escherichia coli, the locations of over one thousand genetic markers have been precisely positioned on its chromosomal map by numerous transduction experiments. In Chapter 3, a mathematical model is constructed based on all known biological processes involved in transduction, so that co-transduction frequencies can be converted into distances between genetic markers. This traditional map provides the basis for the eventual construction of the physical map.

Most of the metabolic pathways involve a collection of different enzymes.  To simplify experimental studies, each enzymatic reaction is usually measured separately by mixing purified enzyme molecules with substrate and co-enzyme. Even in such isolated measurements, the analytical equations are still too complicated. In Chapter 4, simple enzyme kinetic equations are derived as a set of non-linear ordinary differential equations.  Various steps of further simplification are illustrated in detail. One of the original sets of experimental measurements by Michaelis and Menten is then compared with these theoretical results.

For 40 years, many three dimensional structures of proteins have been determined by X-ray diffraction studies, and to a lesser extent by nuclear magnetic resonance studies. Atomic coordinates of non-hydrogen atoms are available from Protein Data Bank. Chapter 5 explains how these coordinates can be used to calculate bond distances, bond angles, f and y angles of the protein backbone. Furthermore, hydrogen atoms can be theoretically positioned into these three dimensional structures, in order to analyze steric hindrance. Sterically hindered (f,y) combinations should be avoided in future refinements.

In Chapter 6, one of the predictive methods of estimating protein backbone (f,y) angles based on the local amino acid sequence and side chain c angles based on steric hindrance is illustrated. This approach is then applied to predict three-dimensional structures of short peptide hormones, and of antibody complementarity determining regions. Other analytical methods of predicting protein tertiary structures based on their primary sequences should be developed, since many amino acid sequences become available from various genome projects.

The backbone structure of nucleic acids is much more complicated. Chapter 7 explains that each nucleotide unit has five single bonds with rotational degree of freedom and some flexibility of its ribose ring. Thus, a minimum of a five-dimensional space will be required for detailed analysis of its possible configurations without steric hindrance. Attempts to simply this complicated problem with two virtual bonds are discussed. Due to its complexity, analytic studies of three-dimensional structures of nucleotides and nucleic acids will be a very challenging problem for both molecular biologists and applied mathematicians.

In Chapter 8, the classical problem of DNA double helix is discussed. In order to analyze the X-ray diffraction pictures of DNA fibers at 92% and 66% relative humidity, we have to thoroughly understand some of the basic properties of Fourier transform, Bessel functions, complex variables, etc. It is hoped that this chapter can eliminate the lack of knowledge of these mathematical methods among molecular biologists. Therefore, in the Appendix, some of the basic tools of applied mathematics are summarized.

Finally, models are by definition simplified versions of the real molecular biology processes. Therefore, constant improvements are essential as more experimental data become available. Readers are encouraged to make such modifications, as well as to propose their own analytical approaches to the problems mentioned in this book and other important molecular biology problems.