Unification of Four Approaches to the Genetic Code

A proposal unifying four approaches to genetic code is discussed. The first approach is introduced by myself and is geometric: genetic code is interpreted as an imbedding of the aminoacid space to DNA space possessing a fiber bundle like structure with DNAs coding for a given aminoacid forming a discrete fiber with a varying number of points. Also Khrennikov has proposed an analogous approach based on the identification of DNAs coding for a given aminoacid as an orbit a discrete flow defined by iteration of a map of DNA space to itself. Much later (2014) I have introduced a variant of this scenario in which the fiber space structure is by assigning aminoacids to the 20 vertices of icosahedron. This model allows to understand the degeneracies of genetic code group theoretically. Second approach starts from the 5-adic approach of Dragovich and Dragovich. Codons are labelled by 5-adic integers n which have no non-vanishing 5-digits so that the n is in the range [31, 124]. The number of primes in the range [31, 124] is 20. This suggests the labelling of aminoacids by these primes. This inspires an additional condition on the geometric code: if possible, one of the integers n projected to p equals to p(n). This condition fails only for the primes 53,79,101,103 for which some of 5-digits vanishing in 5-ary expansion. The third approach relies on the generalization of the basic idea of the so called divisor code proposed by Khrennikov and Nilsson. The requirement is that the number of factors for integer n labelling one of DNAs, call it nd coding for a given aminoacid is the total number of codons coding for the aminoacid, its degeneracy. Therefore a given aminoacid labelled by prime p with no non-vanishing 5-digits is coded by DNAs labelled by p itself and by nd. A group theoretic and physical interpretation for the origin of the divisor code is proposed. The fourth approach is a modification of the earlier 4-adic number theoretic thermodynamics approach of Pitkänen. 1. 5-adic thermodynamics involving a maximization of number theoretic negentropy Np(n) = −Sp(n) > 0(!) as a function of p-adic prime p labelling aminoacids assigns a unique prime to the codon. If no prime in the range divides Sp, the codon is identified as a stopping codon. 2. The number theoretic thermodynamics is assigned with the partitions P of the integer n2) determined by the first two letters of the codon (16 integers belonging to the range [6, 24]). The integer valued number theoretic Hamiltonian h(P ) ∈ Z25 appearing in the Boltzmann weight 5 )/T5 is assumed to depend on the number r of summands for the partition only. h(r) is assumed to be tailored by evolution so that it reproduces the code. 3. The effect of the third nucleotide is described in terms of 5-adic temperature T5 = 1/n, n ∈ [0, 24]: the variation of T5 explains the existence of variants of genetic code and its temporal variation the observed context sensitivity of the codon-aminoacid correspondence for some variants of the code. A numerical calculation scanning over N ∼ 10 candidates for h(r) allows only 11 Hamiltonians and with single additional symmetry inspired condition there are 2 solutions which differ only for 5 largest values of r. Due to the limited computational resources available only 24 percent of the available candidates have been scanned and the naive expectation is that the total number of Hamiltonians is about about 45 unless one poses additional conditions. The problem of the number theoretic models is that they do not predict but only reproduce. This is in sharp contrst to the model based on dark proton sequences, which leads to a radically new vision about the evolution of prebiotic life and to the vision about how immune system and genetic code evolved and what is the meaning of the genetic code.