Hadamard matrices of genetic code and trigonometric functions

Abstract

Algebraic theory of coding is one of the modern fields of applications of algebra. Genetic matrices and algebraic biology have been the latest advances in further understanding of the patterns and rules of genetic code. Genetics code is encoded in combinations of the four nucleotides (A, C, G, T) found in DNA and then RNA. DNA defines the structure and function of an organism and contains complete genetic information. DNA paired bases of (A, C, G, T) form a geometric curve of double helix, define the 64 standard genetic triplets, and further degenerate 64 genetic codons into 20 amino acids. In trigonometry, four basic trigonometric functions (sin x, tan x, cos x, cot x) provided bases for Fourier analysis to encode signal information. In this paper, we use these 4 paired bases of trigonometric functions (sin x, tan x, cos x, and cot x) to generate 64 trigonometric triplets similar to 64 standard genetic code, further exam these 64 trigonometric functions and obtained 20 trigonometric triplets similar to 20 amino acids. This parallel shows a similarity connection between universal genetic codes and the universality of trigonometric functions. This connection may provide a bridge to further uncover patterns of genetic code. This demonstrates that matrix algebra is one of promising instruments and of adequate languages in bioinformatics and algebraic biology.

Keywords

• Hadamard matrix
• trigonometric functions
• genetic code
• algebraic biology

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