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THE SECRETS OF LIFE: A MATHEMATICIAN'S INTRODUCTION TO MOLECULAR BIOLOGY 5 The amino acid sequence of a protein causes it to fold into the particular three-dimensional shape having the lowest energy. This gives the protein its specific biochemical properties, that is, its function. Typically, the shape of a protein is quite robust. If the protein is heated, it will be denatured (that is, lose its three-dimensional structure), but it will often reassume that structure (refold) when cooled. Predicting the folded structure of a protein from the amino acid sequence remains an extremely challenging problem in mathematical optimization. The challenge is created by the combinatorial explosion of plausible shapes, each of which represents a local minimum of a complicated nonconvex function of which the global minimum is sought. CLASSICAL GENETICS The second major approach to studying biological function has been genetics. Whereas biochemists try to study one single component purified away from the organism, geneticists study mutant organisms that are intact except for a single component. Thus a biochemist might study an organism's ability to metabolize sugar by finding mutants that have lost the ability to grow using sugar as a food source. Genetics can be traced back to the pioneering experiments of Gregor Mendel in 1865. These key experiments elegantly illustrate the role of theory and abstraction in biology. For his experiments, Mendel started with pure breeding strains of peasâthat is, ones for which all offspring, generation after generation, consistently show a trait of interest. This choice was key to interpreting the data. One of the traits that he studied was whether the pea made round or wrinkled seeds. Starting with pure breeding round and wrinkled strains, Mendel made a controlled cross to produce an F1, generation. (The ith generation of the cross is denoted Fi.) Mendel noted that all of the F1 generation consisted of round peas; the wrinkled trait had completely vanished. However, when Mendel crossed these F1 peas back to the pure breeding wrinkled parent, the wrinkled trait reappeared: of the second generation, approximately half were round and half were wrinkled. Moreover, when Mendel crossed the F1 peas to themselves, he found that
THE SECRETS OF LIFE: A MATHEMATICIAN'S INTRODUCTION TO MOLECULAR BIOLOGY 6 the second generation showed 75 percent round and 25 percent wrinkled (Figure 1.4). Figure 1.4 Mendel's crosses between pure breeding peas with round and wrinkled seeds revealed the telltale binomial ratio 1:2:1 in the second generation that led Mendel to infer the existence of discrete particles of inheritance. On the basis of these and other experiments, Mendel hypothesized that traits such as roundness are affected by discrete factorsâwhich today we call genes. In particular, Mendel suggested the following: ⢠Each organism inherits two copies of a gene, one from each parent. Each parent passes on one of the two copies, chosen at random, to each offspring. (These important postulates are called Mendel's First Law of Inheritance.) ⢠Genes can occur in alternative forms, called alleles. For example, the gene affecting seed shape occurs in one form (allele A) causing roundness and one form (allele a) causing wrinkledness. ⢠The pure breeding round and wrinkled plants carried two copies of the same allele, AA and aa, respectively. Individuals carrying two copies of the same gene are called homozygotes. The F1 generation consists of individuals with genotype Aa, with the round trait dominant over the wrinkled trait. Such individuals are called heterozygotes.
