Mathematical methods that illuminate fundamental problems related to the genetic code and bioinformatics

Mathematics of Bioinformatics: Theory, Practice, and Applications provides a comprehensive blueprint for connecting and integrating information derived from mathematical methods and applying it to the understanding of biological sequences, structures, and networks. It offers valuable knowledge about mathematical tools, phenomenological results, and interdisciplinary connections in the fields of molecular genetics, bioinformatics, and informatics.

Each chapter is divided into sections based on bioinformatics topics and related mathematical theory and methods. Each topic is comprised of an introduction to the biological problems in bioinformatics; a presentation of topics in mathematical theory and methods relevant to the problems; and an integrative overview that draws the connections and interfaces between the problems, theory, methods, and applications. This practical resource:

• Covers genetic codes, sequences, structures, functions, biological networks/systems, and interfaces with mathematics, making connections between mathematics and bioinformatics for the bioinformatics specialist
• Provides integrative models for potential simulations, modeling, and implementation utilizing algorithms and analysis for the computer scientist
• Details recent research covering other branches of mathematics such as linear algebra, topology, differential geometry, fractals, and chaos theory that have found useful applications in bioinformatics
• Emphasizes applications of mathematics in bioinformatics while eschewing mathematical proofs and deep theories

Mathematics of Bioinformatics is intended for scientists, researchers, and upper-level undergraduate and graduate students in bioinformatics, mathematics, computer informatics, theoretical biology, mathematical biology, and biotechnology who seek information on the possibilities and challenges of interface between mathematics and bioinformatics. Readers with a foundation in calculus can also adapt to the mathematical topics introduced throughout.

Matthew He, PhD, is Full Professor and Director of the Division of Math, Science, and Technology of Nova Southeastern University, Florida. He is Full Professor and Grand PhD from the World Information Distributed University, Belgium, since 2004. Dr. He has published more than 100 research papers in mathematics, computer science, information theory, and bioinformatics, and is an editor of both International Journal of Biological Systems and International Journal of Cognitive Informatics and Natural Intelligence.

Sergey Petoukhov, PhD, is a chief scientist of the Department of Biomechanics, Mechanical Engineering Research Institute of the Russian Academy of Sciences, Moscow, as well as Full Professor and Grand PhD from the World Information Distributed University. He has published more than 150 research papers in biomechanics, bioinformatics, mathematical and theoretical biology, the theory of symmetries and its applications, and mathematics.

Mathematics of Bioinformatics

Theory, Methods and ApplicationsBy Matthew He Sergey Petoukhov

John Wiley & Sons

Copyright © 2011 John Wiley & Sons, Inc.
All right reserved.
ISBN: 978-0-470-40443-0

Chapter One

Bioinformatics and Mathematics

Traditionally, the study of biology is from morphology to cytology and then to the atomic and molecular level, from physiology to microscopic regulation, and from phenotype to genotype. The recent development of bioinformatics begins with research on genes and moves to the molecular sequence, then to molecular conformation, from structure to function, from systems biology to network biology, and further investigates the interactions and relationships among, genes, proteins, and structures. This new reverse paradigm sets a theoretical starting point for a biological investigation. It sets a new line of investigation with a unifying principle and uses mathematical tools extensively to clarify the ever-changing phenomena of life quantitatively and analytically.

It is well known that there is more to life than the genomic blueprint of each organism. Life functions within the natural laws that we know and those that we do not know. Life is founded on mathematical patterns of the physical world. Genetics exploits and organizes these patterns. Mathematical regularities are exploited by the organic world at every level of form, structure, pattern, behavior, interaction, and evolution. Essentially all knowledge is intrinsically unified and relies on a small number of natural laws. Mathematics helps us understand how monomers become polymers necessary for the assembly of cells. Mathematics can be used to understand life from the molecular to the biosphere levels, including the origin and evolution of organisms, the nature of genomic blueprints, and the universal genetic code as well as ecological relationships.

Mathematics and biological data have a synergistic relationship. Biological information creates interesting problems, mathematical theory and methods provide models for understanding them, and biology validates the mathematical models. A model is a representation of a real system. Real systems are too complicated, and observation may change the real system. A good system model should be simple, yet powerful enough to capture the behavior of the real system. Models are especially useful in bioinformatics. In this chapter we provide an overview of bioinformatics history, genetic code and mathematics, background mathematics for bioinformatics, and the big picture of bioinformatics–informatics.

1.1 INTRODUCTION

Mendel’s Genetic Experiments and Laws of Heredity The discovery of genetic inheritance by Gregor Mendel back in 1865 was considered as the start of bioinformatics history. He did experiments on the cross-fertilization of different colors of the same species. Mendel’s genetic experiments with pea plants took him eight years (1856–1863). During this time, Mendel grew over 10,000 pea plants, keeping track of progeny number and type. He recorded the data carefully and performed mathematical analysis of the data. Mendel illustrated that the process of inheritance of traits could be explained more easily if it was controlled by factors passed down from generation to generation. He concluded that genes come in pairs. Genes are inherited as distinct units, one from each parent. He also recorded the segregation of parental genes and their appearance in the offspring as dominant or recessive traits. He published his results in 1865. He recognized the mathematical patterns of inheritance from one generation to the next. Mendel’s laws of heredity are usually stated as follows:

• The law of segregation. A gene pair defines each inherited trait. Parental genes are randomly separated by the sex cells, so that sex cells contain only one gene of the pair. Offspring therefore inherit one genetic allele from each parent. • The law of independent assortment. Genes for different traits are sorted from one another in such a way that the inheritance of one trait is not dependent on the inheritance of another. • The law of dominance. An organism with alternate forms of a gene will express the form that is dominant.

In 1900, Mendel’s work was rediscovered independently by DeVries, Correns, and Tschermak, each of whom confirmed Mendel’s discoveries. Mendel’s own method of research is based on the identification of significant variables, isolating their effects, measuring these meticulously, and eventually subjecting the resulting data to mathematical analysis. Thus, his work is connected directly to contemporary theories of mathematics, statistics, and physics.

Origin of Species Charles Darwin published On the Origin of Species by Means of Natural Selection (Darwin, 1859) or “The Preservation of Favored Races in the Struggle for Life.” His key work was that evolution occurs through the selection of inheritance and involves transmissible rather than acquired characteristics between individual members of a species. Darwin’s landmark theory did not specify the means by which characteristics are inherited. The mechanism of heredity had not been determined at that time.

First Genetic Map In 1910, after the rediscovery of Mendel’s work, Thomas Hunt Morgan at Columbia University carried out crossing experiments with the fruit fly (Drosophila melanogaster). He proved that the genes responsible for the appearance of a specific phenotype were located on chromosomes. He also found that genes on the same chromosome do not always assort independently. Furthermore, he suggested that the strength of linkage between genes depended on the distance between them on the chromosome. That is, the closer two genes lie to each other on a chromosome, the greater the chance that they will be inherited together. Similarly, the farther away they are from each other, the greater the chance of that they will be separated in the process of crossing over. The genes are separated when a crossover takes place in the distance between the two genes during cell division. Morgan’s experiments also lead to Drosophila‘s unusual position as, to this day, one of the best studied organisms and most useful tools in genetic research. In 1911, Alfred Sturtevant, then an undergraduate researcher in the laboratory of Thomas Hunt Morgan, mapped the locations of the fruit fly genes, creating the first genetic map ever made.

Transposable Genetic Elements In 1944, Barbara McClintock discovered that genes can move on a chromosome and can jump from one chromosome to another. She studied the inheritance of color and pigment distribution in corn kernels at the Carnegie Institution Department of Genetics in Cold Spring Harbor, New York. At age 81 she was awarded a Nobel prize. It is believed that transposons may be linked to such genetic disorders as hemophilia, leukemia, and breast cancer; and transposons may have played a crucial role in evolution.

DNA Double Helix In 1953, James Watson and Francis Crick proposed a double-helix model of DNA. DNA is made of three basic components: a sugar, an acid, and an organic “base.” The base was always one of the four nucleotides: adenine (A), cytosine (C), guanine (G), or thymine (T). These four different bases are categorized in two groups: purines (adenine and guanine) and pyrimidines (thymine and cytosine). In 1950, Erwin Chargaff found that the amounts of adenine (A) and thymine (T) in DNA are about the same, as are the amounts of guanine (G) and cytosine (C). These relationships later became known as “Chargaff’s rules” and led to much speculation about the three-dimensional structure that DNA would have. Rosalind Franklin, a British chemist, used the x-ray diffraction technique to capture the first high-quality images of the DNA molecule. Franklin’s colleague Maurice Wilkins showed the pictures to James Watson, an American zoologist, who had been working with Francis Crick, a British biophysicist, on the structure of the DNA molecule. These pictures gave Watson and Crick enough information to propose in 1953 a double-stranded, helical, complementary, antiparallel model for DNA. Crick, Watson, and Wilkins shared the 1962 Nobel Prize in Physiology or Medicine for the discovery that the DNA molecule has a double-helical structure. Rosalind Franklin, whose images of DNA helped lead to the discovery, died of cancer in 1958 and, under Nobel rules, was not eligible for the prize. In 1957, Francis Crick and George Gamov worked out the “central dogma,” explaining how DNA functions to make protein. Their sequence hypothesis posited that the DNA sequence specifies the amino acid sequence in a protein. They also suggested that genetic information flows only in one direction, from DNA to messenger RNA to protein, the central concept of the central dogma.

Genetic Code (see Appendix A) The genetic code was finally “cracked” in 1966. Marshall Nirenberg, Heinrich Mathaei, and Severo Ochoa demonstrated that a sequence of three nucleotide bases, a codon or triplet, determines each of the 20 amino acids found in nature. This means that there are 64 possible combinations (4³ = 64) for 20 amino acids. They formed synthetic messenger ribonucleic acid (mRNA) by mixing the nucleotides of RNA with a special enzyme called polynucleotide phosphorylase. This resulted in the formation of a single-stranded RNA in this reaction. The question was how these 64 genetic codes could code for 20 different amino acids. Nirenberg and Matthaei synthesized poly(U) by reacting only uracil nucleotides with the RNA-synthesizing enzyme, producing UUUU. They mixed this poly(U) with the protein-synthesizing machinery of Escherichia coli in vitro and observed the formation of a protein. This protein turned out to be a polypeptide of phenylalanine. They showed that a triplet of uracil must code for phenylalanine. Philip Leder and Nirenberg found an even better experimental protocol to solve this fundamental problem. By 1965 the genetic code was solved almost completely. They found that the “extra” codons are merely redundant: Some amino acids have one or two codons, some have four, and some have six. Three codons (called stop codons) serve as stop signs for RNA-synthesizing proteins.

First Recombinant DNA Molecules In 1972, Paul Berg of Stanford University created the first recombinant DNA molecules by combining the DNA of two different organisms. He used a restriction enzyme to isolate a gene from a human-cancer-causing monkey virus. Then he used lipase to join the section of virus DNA with a molecule of DNA from the bacterial virus lambda, creating the first recombinant DNA molecule. He realized the risks of his experiment and terminated it temporarily before the recombinant DNA molecule was added to E. coli, where it would have quickly been reproduced. He proposed a one-year moratorium on recombinant DNA studies while safety issues were addressed. Berg later resumed his studies of recombinant DNA techniques and was awarded the 1980 Nobel Prize in Chemistry. His experiments paved the road for the field of genetic engineering and the modern biotechnology industry.

DNA Sequencing and Database In early 1974, Frederick Sanger from the UK Medical Research Council was first to invent DNA-sequencing techniques. During his experiments to uncover the amino acids in bovine insulin, he developed the basics of modern sequencing methods. Sanger’s approach involved copying DNA strands, which would show the location of the nucleotides in the strands. To apply Sanger’s approach, scientists had to analyze the composite collections of DNA pieces detected from four test tubes, one for each of the nucleotides found in DNA (adenosine, cytosine, thymidine, guanine). Then they needed to be arranged in the correct order. This technique is very slow and tedious. It takes many years to sequence only a few million letters in a string of DNA. Almost simultaneously, the American scientists Alan Maxam and Walter Gilbert were creating a different method called the cleavage method. The base for virtually all DNA sequencing was the dideoxychain-terminating reaction developed by Sanger.

In 1978, David Botstein developed restriction-fragment-length polymorphisms. Individual human beings differ one base pair in every 500 nucleotides or so. The most interesting variations for geneticists are those that are recognized by certain enzymes called restriction enzymes. Each of these enzymes cuts DNA only in the presence of a specific sequence (e.g., GAATTC in the case of the restriction enzyme EcoR1). This sequence is called a restriction site. The enzyme will bypass the region if it has mutated to GACTTC. Thus, when a specific restriction enzyme cuts the DNA of different people, it may produce fragments of different lengths. These DNA fragments can be separated according to size by making them move through a porous gel in an electric field. Since the smaller fragments move more rapidly than the larger ones, their sizes can be determined by examining their positions in the gel. Variations in their lengths are called restriction-fragment-length polymorphisms.

In 1980, Kary Mullis invented polymerase chain reaction (PCR), a method for multiplying DNA sequences in vitro. The purpose of PCR is to make a huge number of copies of a specific DNA fragment, such as a gene. Use of thermostable polymerase allows the dissociation of newly formed complementary DNA and subsequent annealing or hybridization of the primers to the target sequence with a minimal loss of enzymatic activity. PCR may be necessary to receive enough starting template for instance sequencing.

In 1986, scientists presented a means of detecting ddNTPs with fluorescent tags, which required only a single test tube instead of four. As a result of this discovery, the time required to process a given batch of DNA was reduced by one-fourth. The amount of sequenced base pairs increased rapidly from there on.

Established in 1988 as a national resource for molecular biology information, the National Center for Biotechnology Information (NCBI) carries out diverse responsibilities. NCBI creates public databases, conducts research in computational biology, develops software tools for analyzing genome data, and disseminates biomedical information: all for a better understanding of molecular processes affecting human health and disease. NCBI conducts research on fundamental biomedical problems at the molecular level using mathematical and computational methods.

The European Bioinformatics Institute (EBI) is a nonprofit academic organization that forms part of the European Molecular Biology Laboratory (EMBL). The roots of the EBI lie in the EMBL Nucleotide Sequence Data Library, which was established in 1980 at the EMBL laboratories in Heidelberg, Germany and was the world’s first nucleotide sequence database. The original goal was to establish a central computer database of DNA sequences rather than having scientists submit sequences to journals. What began as a modest task of abstracting information from literature soon became a major database activity with direct electronic submissions of data and the need for a highly skilled informatics staff. The task grew in scale with the start of the genome projects, and grew in visibility as the data became relevant to research in the commercial sector. It became apparent that the EMBL Nucleotide Sequence Data Library needed better financial security to ensure its long-term viability and to cope with the sheer scale of the task.

Human Genome Project In 1990, the U.S. Human Genome Project started as a 15-year effort coordinated by the U.S. Department of Energy and the National Institutes of Health. The project originally was planned to last 15 years, but rapid technological advances accelerated the expected completion date to 2003. Project goals were to:

• Identify all the genes in human DNA • Determine the sequences of the 3 billion chemical base pairs that make up human DNA • Store this information in databases • Improve tools for data analysis • Transfer related technologies to the private sector • Address the ethical, legal, and social issues (ELSIs) that may arise from the project

In 1991, working with Nobel laureate Hamilton Smith, Venter’s genomic research project (TIGR) created the shotgunning method. At first the method was controversial among Venter’s colleagues, who called it crude and inaccurate. However, Venter cross-checked his results by sequencing the genes in both directions, achieving a level of accuracy that greatly impressed his initial sceptical rivals. Within a year, TIGR published the entire genome of Haemophilus influenzae, a bacterium with nearly 2 million nucleotides.

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