Genetics and mathematics

Summary: Bioinformatics and probability theory come into play in the study of genetics.

Issues related to genetics are no longer exclusively discussed in academic circles. The lay community every day accesses a large amount of information through the mass communication vehicles that enable the socialization of knowledge related to heredity and biotechnology. Paternity tests, transgenic plants, early diagnoses in medicine, gene therapy, and cloning are no longer exclusive subjects of specialized research centers and can be easily researched in the media and found in movies, cartoons, and on the Internet. These are examples of how closely aligned this area of science is to modern society and how broad the possibilities are for development. Mathematical tools are essential for the analysis and interpretation of data related to these genetic processes that otherwise would become empty of real meaning. From the simple knowledge of probability to the most powerful algorithms associated with genetic engineering techniques, probability is necessary to elucidate the most difficult questions surrounding the genetics field. Sharon Grossman is one mathematician who has notably contributed to genetics research through her investigations of gene groupings based on geographic location.

Genetics is the field of biology that studies the chemical nature of hereditary material and the mechanism responsible to transfer information contained in genes. In general, reproduction is constituted by a series of events that result in a randomized combination of gametes. This process involves the mixing of thousands of information packets and results in the production of a new living being.

Early Findings

The first steps of genetics were performed by Austrian Gregor Mendel (1822–1884), who, for many years, crossed varieties of peas. After obtaining numerous generations of these plants, he observed differences in the types of progeny formed and identified the proportion of each of these features in future generations. His main findings showed that specific factors were transmitted by parents to offspring. He also found that these factors occur in pairs and that their descendants receive one from each parent. Crosses made with peas (called “hybrids”) had particular characteristics, like seed color. By calculating their frequencies, Mendel realized that the prevalence of these factors was different in several generations. Some manifest themselves only when appearing in double dose (recessive), while others in a single dose determined the characteristic (dominant). These findings served as the basis for developing laws on inheritance, which came to be called the “first and second laws of Mendel.”

Genetic Probability

Probabilities are used to express the chance of occurrence of an event. They represents a possibility, not a conviction. The probabilities can be expressed in several ways, including fractions, percentages, and decimals. For example, the chance of occurrence of a biological event can be expressed as “50%,” “0.50,” or “1/2.” Many genetics calculations are solved using probability. Mendel used mathematical rules previously used for common events, such as a coin toss (individual events), or combined events, such as the simultaneous release of multiple dice.

Genotype is the set of genes from one living being, the frequency of these genes, and can be calculated mathematically. The calculations performed in the theory of probability do not determine the appearance of a particular genotype—they merely represent the chance this event will occur. In practical terms, genetic calculations allow one to determine the probability, for example, of two individuals with dark eyes to conceive a child with blue eyes (a recessive gene). This event is possible if both parents are hybrids, in which case the probability in each pregnancy is 25%.

The application of rigorous scientific method and careful statistical research of some characteristics led Mendel to conclusions that still underlie modern genetics in the early twenty-first century. Not until in 1900 could the work of three independent researchers—Hugo de Vries, Karl Correns, and Erich Tschermak—show that Mendel’s conclusions were correct.

Modern Genetics Research

For a long time it was believed that protein was the molecule that contained genetic information. Biochemical studies allowed the identification of a molecule able to replicate and thereby allow a flow of identification information: deoxyribonucleic acid (DNA)—the molecule associated with heredity. In 1953, James Watson and Francis Crick published in Nature the model of the DNA molecule. Understanding the complex spatial geometry of DNA allowed researchers in the early 1960s to prove that the code was formed by groups of three nucleotides that were repeated in complementary sequences. It was noticed that sequences of the DNA molecule were able to be expressed as proteins with the participation of another nucleic acid: ribonucleic acid (RNA). Studies of the most primitive life forms, like bacteria, also led to knowledge related to peculiarities of DNA activity, as well as its transmission and their biochemical behavior.

The challenge became the elucidation of the genome (the entire set of genetic information that is found in the chromosomes) from a living organism. After advancing with some simple life forms, such as bacteria and protozoa, the Human Genome Project (1988–2003) arose. International cooperation efforts were necessary to decipher the sequence of 3 billion base pairs of DNA subunits found in human chromosomes. Powerful computer programs and the use of combinatorial analysis revealed that most of the DNA molecule is not involved in protein-coding. It is now known, however, that the role of this DNA is very significant, especially for matters pertaining to evolution, and it is responsible for many adaptive differences between species.

Genetic Variability

The prevalence of certain genes in a population depends on how the expression of a particular feature is selected by the environment and is related to the presence of other genetic variation factors, such as genetic mutations, numbers of crosses, or natural events that abruptly decrease the frequency of certain genes in a population (for example, earthquakes, fires, or floods).

How is it possible to evaluate this natural dynamic that sometimes takes decades or even centuries to occur? Since a group within the set of genes undergoes a random process of transmission, it cannot be adequately studied without resorting to mathematical tools to assess the frequency of certain genes in a population and the possible consequences of this variability for that group. Wild populations (animals, plants, or, specifically, humans) are subject to phenomena—such as gene recombination, mutation, and gene conversion, which is the change of position of genes within a chromosome—that lead to the emergence of genetic variability. Genetic mathematics aims to understand how genetic changes occur for individuals both within species and over time.

Several phenomena are responsible for genetic variability. Crossing-over, for example, is a phenomenon in which parts of chromosomes are broken and glued in different positions, generating a larger mix of information and expression of phenotypes (physical or physiological), which results in an increased possibility of adapting to the environment in which the individual belongs. Random events observed in gene transfer result in the formation of functional characteristics and patterns that may often cause trouble and injury, but that is partly responsible for the possibility of evolution.

Bioinformatics

Genetics is an area of study that uses the theories of probability and the handling of large volumes of data. The difficulties in performing complex calculations—far more advanced than the calculations made by Mendel—necessitated the use of information technology in studies of biological and genetic research. Bioinformatics is the application of computer systems in the processing of biological and biomedical data. It is an interdisciplinary science that aims to develop and apply computational techniques to study genetics, molecular biology, and biochemistry. Without this tool, it would be impossible to perform thousands of mathematical operations in real time.

In bioinformatics, gene sequences are analyzed and stored in databases, manipulated, and analyzed using specific software. Databases allow scientists to get information from other laboratories and also to share the genetic sequences. Despite the efforts of international collaboration in this area, the patenting of genes clashes science with ethical issues regarding the detention of the natural heritage of knowledge as private property. Laws regarding other issues related to cloning and gene manipulation vary according to country.

Genetic engineering uses principles formulated many years ago. The development of refined methods using molecular biology techniques allowed the manipulation of genetic material, known as “recombinant DNA technology” or “genetic engineering.” Once DNA fingerprinting had become associated with the identification of individuals, great hopes arose regarding the possibility of isolating and cloning genes to replace defective genes as therapy.

Bibliography

“Genetics Home Reference: Your Guide to Understanding Genetic Conditions.” U.S. National Library of Medicine. http://ghr.nlm.nih.gov.

Lachowicz, M., and J. Miekisz, eds. From Genetics to Mathematics. Vol. 79 of Series on Advances in Mathematics for Applied Sciences. Hackensack, NJ: World Scientific Publishing Company, 2009.

“Learn. Genetics: Genetics Science Learning Center.” University of Utah. http://learn.genetics.utah.edu.