p-Adic mathematics and theoretical biology.

The principal objective of this article is a brief overview of the main parts of p-adic mathematics, which have already had valuable applications and may have a significant impact in the near future on the further development of some fields of theoretical and mathematical biology. In particular, we present the basics of ultrametrics, p-adic numbers and p-adic analysis, as well as insight into their applications for modeling some cognitive processes, genetic code and protein dynamics. We also argue that ultrametric concepts and p-adic mathematics are natural tools for the viable description of biological systems and phenomena with a hierarchical structure.

p-Adic hierarchical properties of the genetic code.

In this article, we consider p-adic modeling of the standard genetic code and the vertebrate mitochondrial one. To this end, we use 5-adic and 2-adic distance as a mathematical tool to describe closeness (nearness, similarity) between codons as elements of a bioinformation space. Codons which are simultaneously at the smallest 5-adic and 2-adic distance code the same (or similar) amino acid or stop signal. The set of codons is presented as an ultrametric tree as well as a fractal and p-adic network. It is shown that genetic code can be treated as sequential translation between genetic languages. This p-adic approach gives possibility to be applied to sequences of nucleotides of an arbitrary finite length. We present an overview of published and some new results on various p-adic properties of the genetic code.