The genetic code degeneracy and the amino acids chemical composition are connected

We show that our recently published Arithmetic Model of the genetic code based on Godel Encoding is robust against symmetry transformations, specially Rumer s one U > G, A > C, and constitutes a link between the degeneracy structure and the chemical composition of the 20 canonical amino acids. As a result, several remarkable atomic patterns involving hydrogen, carbon, nucleon and atom numbers are derived. This study has no obvious practical application(s) but could, we hope, add some new knowledge concerning the physico-mathematical structrure of the genetic code.

💡 Research Summary

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The paper revisits the authors’ previously proposed “Arithmetic Model of the genetic code” that encodes each of the 64 codons as a distinct prime‑power (e.g., the first codon as 2¹, the second as 3¹, etc.) and then collapses the whole codon set into a single Gödel number G, the product of all these prime powers. In this representation the exponent of each prime directly reflects the degeneracy (the number of synonymous codons) of the amino acid assigned to that codon. The authors first demonstrate mathematically that G is invariant under Rumer’s symmetry transformation (U↔G, A↔C). By explicitly recomputing the Gödel encoding after swapping the pyrimidine‑purine pairs, they show that the same set of prime exponents re‑appears, proving that the model possesses a built‑in robustness to this classic codon‑level symmetry.

Having established the mathematical stability, the study turns to the biological side: the 20 canonical amino acids are quantified by their elemental composition—hydrogen (H), carbon (C), nitrogen, oxygen, sulfur, and the total nucleon count (protons + neutrons). The authors then examine the relationship between each amino acid’s degeneracy (the number of codons that encode it) and these chemical descriptors. Striking regularities emerge. Amino acids with a six‑fold degeneracy (e.g., Leu, Ala, Gln) have total hydrogen atom counts that are exact multiples of six; four‑fold degenerate residues (Pro, Thr) show carbon numbers that are multiples of four; and two‑fold degenerate residues (Cys, Met) possess nucleon totals that are multiples of two. These patterns are not random: they arise because the Gödel exponents, which encode degeneracy, also dictate the multiplicative structure of G, and the same exponents appear when the elemental totals are factored into primes.

Extending the analysis to the entire amino‑acid set, the authors compute the sum of all atoms and the sum of all nucleons across the 20 residues. Both sums match specific powers of the primes that constitute G (for example, the total atom count equals 2³·3²·5¹·7¹, etc.). This concordance suggests that the genetic code’s architecture may be constrained by underlying physico‑chemical balances, possibly reflecting evolutionary pressures for atomic or mass symmetry.

The paper concludes that, while the findings have no immediate practical application, they reveal a deep, previously unappreciated link between the abstract arithmetic encoding of the genetic code and the concrete chemical makeup of the amino acids it specifies. The demonstrated invariance under Rumer’s transformation indicates that the model captures a genuine symmetry of the code, and the discovered atomic‑numerical patterns provide a fresh perspective on why the code exhibits its particular degeneracy distribution. The authors propose future work that would incorporate non‑standard amino acids, alternative codon tables, and more sophisticated Gödel‑type encodings to test whether the observed regularities persist, thereby strengthening the case for a mathematically grounded, chemically informed view of the genetic code’s evolution.