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THE SECRETS OF LIFE: A MATHEMATICIAN'S INTRODUCTION TO MOLECULAR BIOLOGY 22 for mathematics, statistics, and computer science in modern molecular biology. COMING ATTRACTIONS The chapters of this book describe important applications of mathematical, statistical, and computational methods to molecular biology. These methods are developing rapidly, and, mainly because of this situation, the presentations in this book are intended to be introductory sketches rather than scholarly reviews. Without claiming to be a complete survey, this book should convey to readers some of the exciting uses of mathematics, statistics, and computing in molecular biology. Other introductions to various aspects of molecular biology can be found in Watson et al. (1994), Streyer (1988), U.S. Department of Energy (1992), Watson et al. (1987), Lewin (1990), and Alberts et al. (1989). Chapter 2 ("Mapping Heredity") describes how statistical models can be used to map the approximate location of genes on chromosomes. Gene mapping was mentioned above for the case of the cystic fibrosis gene. The problem becomes especially challengingâand mathematics plays a bigger roleâwhen the disease does not follow simple Mendelian inheritance patternsâfor example, when it is caused by multiple genes or when the trait is quantitative rather than qualitative in nature. This is an important subject for the Human Genome Project and its applications in modern medical genetics. The next three chapters focus on the analysis of DNA and protein sequences. As new genes are sequenced, they are routinely compared with public databases to look for similarities that might indicate common evolutionary origin, structure, or function. As databases expand at ever-increasing rates, the computational efficiency of such comparisons is crucial. Chapter 3 ("Seeing Conserved Signals") describes combinatorial algorithms for this problem. Because coincidences abound in such comparisons, careful statistical analysis is needed. Chapter 4 ("Hearing Distant Echoes") discusses the application of extremal statistics to sequence similarity. For closely related sequences, sequence comparison also sheds light on the process of evolution. Chapter 5 ("Calibrating the Clock") discusses the applications
THE SECRETS OF LIFE: A MATHEMATICIAN'S INTRODUCTION TO MOLECULAR BIOLOGY 23 of stochastic processes to such evolutionary analysis. The discovery and reading of genetic sequences have breathed new life into the study of the stochastic processes of evolution. The chapter focuses on one of the most exciting new tools, the use of the coalescent to estimate times to the most recent common ancestor. Geometric methods applied to DNA structure and function are the focus of the next three chapters. Watson and Crick's famous DNA double helix can be thought of as local geometrical structure. There is also much interesting geometry in the more global structure of DNA molecules. Chapter 6 ("Winding the Double Helix") uses methods from geometry to describe the coiling and packing of chromosomes. The chapter describes the supercoiling of the double helix, in terms of key geometric quantitiesâlink, twist, and writheâthat are related by a fundamental theorem. Chapter 7 ("Unwinding the Double Helix") employs differential mechanics to study how stresses on a DNA molecule cause it to unwind in certain areas, thereby allowing access by key enzymes needed for gene expression. Chapter 8 ("Lifting the Curtain") uses topology to infer the mechanism of enzymes that recombine DNA strands, providing a glimpse of details that cannot be seen via experiment. Finally, Chapter 9 ("Folding the Sheets") discusses one of the hardest open questions in computational biology: the protein-folding problem, which concerns predicting the three-dimensional structure of a protein on the basis of the sequence of its amino acids. Probably no simple solution will ever be given for this central problem, but many useful and interesting approximate approaches have been developed. The concluding chapter surveys various computational approaches for structure prediction. Together, these chapters provide glimpses of the roles of mathematics, statistics, and computing in some of the most exciting and dynamic areas of molecular biology. If this book tempts some mathematicians, statisticians, and computational scientists to learn more about and to contribute to molecular biology, it will have accomplished one of its goals. Its two other goals are to encourage molecular biologists to be more cognizant of the importance of the mathematical and computational sciences in molecular biology and to encourage scientifically literate people to be aware of the increasing impact of both molecular biology and mathematical and computational sciences on their
