What follows is a vastly oversimplified look at molecular biology and genetics. However, without basic knowledge of the subject area in which mathematics is being used, it is impossible either to use it effectively or to follow what has been done and is being done. The biological situation is rather different for bacteria and viruses than it is for higher organisms. Also, the situation for plants is quite different from sexually reproducing animals, which is what I will primarily be concerned with.
The genetic material for humans is located in the nucleus of cells. This nucleus contains structures called chromosomes. There are 23 pairs of chromosomes in humans (46 chromosomes in all). Twenty-two pairs are matched or homologous pairs, which are inherited one from each parent. These chromosomes are known as autosomes. The members of the extra pair of chromosomes are known as sex chromosomes. In humans, if one has two X chromosomes one is female, while if one has an X and Y chromosome one is male. One gets a sex chromosome from one's mother (one of her two X chromosomes), and one from one's father (either an X or a Y chromosome) with the result that one has a 50-50 chance of a child's being a boy or a girl. Because of this special mode of inheritance, one refers to sex-linked traits if the mechanism of inheritance involves the X or Y chromosome. Most human cells have pairs of chromosomes and are referred to as diploid. However, the sex cells, egg and sperm, also known as gametes, have only a single set of chromosomes and are known as haploid. A zygote results from the fusing of a sperm and an egg and is diploid. Mitosis is a process in which the nucleus of a cell produces a pair of nuclei that contain identical copies of chromosomes to each other and to the nucleus that produced them. Meiosis is a special form of nuclear cell division which results in the formation of haploid gametes. During the complex processes of mitosis or meiosis (and in other situations) sometimes parts of the chromosomes are inverted, deleted, duplicated, damaged, or moved to other locations on the chromosomes. The reason for this may be related to radiation, chemicals, heat, etc. Such changes may or may not affect a gene section of the chromosome. Such changes are referred to as mutations.
The chromosomes in a human consist of long strands of DNA (deoxyribonucleic acid) which are compacted in the chromosome. Oswald Avery was a pioneer in conducting experiments connecting DNA with the chemical basis of heredity. Armed with Erwin Chargaff's work on the chemical components of DNA, and x-ray crystallographic insights from Rosalind Franklin, Francis Crick and James Watson proposed that DNA molecules are double helices. Our understanding of the function of all the DNA in a chromosome is rather rudimentary, despite spectacular progress.
A cell showing the chromosomes within its nucleus, as well as some detail of the chromosomes and the DNA which make them up.