Mathematics and Codons

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The book consists of six chapters: - Elementary Number Theory, Algebra, and Linear Algebra - Representations of Codons and the Protein Code - Differential Geometry of Space Curves and Supplements to Groups - Elementary Protein Structure and Similar Codes - Linear Systems Theory - Repetition Codes and Nonlinear Systems. The first chapter contains basic mathematics needed for the rest of the book, that is elementary number theory, algebra, i.e. groups, rings, and in particular finite fields, as well as linear algebra. This may serve as an introduction to those readers not acquainted with these mathematical topics. Within Chapter 2, some introductory remarks on Codons and the protein code are given, while in Section 2 the definition, representation, and geometric aspects of Codons are presented. Thereby, 3-Codons are defined as three-dimensional vectors over the field F(4). Section 3 consists of the assignment to amino acids that rely on the definition of Codon multiplets. By number theoretical and information theoretical methods, these numbers are derived. Chapter 3 gives the mathematical basics relevant for Chapter 4, that are space curves in the sense of differential geometry and motions. For the latter we recover the icosahedron group over the field F(4) and discuss the helix vector. Finally some supplements to groups are given. Elementary protein structure is the content of Chapter 4. While primary, tertiary, and quaternary structure is treated shortly, emphasis is put on secondary structure. For two proteins, namely the cro- and cii-protein of the enterobacteria lambda we calculate the localization of the secondary structures. Finally, similar protein codes are investigated. Linear systems theory is treated in the fifth chapter. It is shown, that there exist phenomena like the nonlocal behaviour of the solutions, period doubling, and self-replication obtained by employing linear equation over finite fields, which usually show up in conjunction with nonlinear equations over the field of real numbers, only. The last chapter treats repetition codes as well as nonlinear systems over finite fields.