Triply Stochastic Cubes Associated with Genetic Code Numerical Mappings

Knowledge about genetic coding systems are useful for computer science, engineering and education. In this paper we derive triply stochastic cubes associated with the triplet genetic code numerical mappings. We also demonstrate the symmetrical patterns between the entries of the cubes and DNA molar concentration accumulation via an arithmetic sequence. The stochastic cubes based on genetic code were derived by using three kinds of chemically determined equivalences. We have shown that at each stage (Nth step) of matrix evolution, hydrogen bonds expansion is triply stochastic and its accumulation is governed by an arithmetic sequence with a common difference of total number of hydrogen bonds of 5N; the pyrimidines/purines ring expansion is triply stochastic and its accumulation is governed by an arithmetic sequence with a common difference of total number of rings of 3N; and the amino-mutating absence/present expansion is also triply stochastic and its accumulation is governed by an arithmetic sequence with a common difference of total number of amino-mutating of 1N. Data about the genetic stochastic matrices/cubes associated with the genetic codes can lead to new understanding of genetic code systems, to new effective algorithms of information processing which has a perspective to be applied for modeling mutual communication among different parts of the genetic ensemble.