The Mathematical Structure of the Genetic Code

Abstract In this chapter a new mathematical theory of the genetic code is presented, based on a particular kind of number representation: a non-power binary representation. A mathematical model is constructed that allows the description of many known properties of the genetic code, such as the degeneracy distribution and the specific codon-amino acid assignation, and also of some new properties such as, for example, palindromic symmetry (a degeneracy preserving transformation), which is shown to be the highest level of a series of hierarchical symmetries. The role of chemical dichotomy classes, which varies between purine-pyrimidine, amino-keto, and strong-weak following the position of the bases in the codon frame, is shown. A new characterization of codons, obtained through the parity of the corresponding binary strings in the mathematical model, together with the associated symbolic structure acting on the codon space, is also illustrated. Furthermore, it is shown that Rumer's classes (or degeneracy classes) can be obtained symbolically from the two first letters of a codon by means of an operation, which is identical to that of parity determination from a structural point of view. On this basis, the existence of a third dichotomy class sharing the former properties can be hypothesized. Two main facts related to this new theory are also discussed: first, the intrinsic parity of codons in the number representation strongly suggests the existence of error detection/correction mechanisms based on such coding. These suggested mechanisms should work on the basis of the same general principles used in manmade systems for the transmission of digital data, like those associated with CDs, DVDs, wireless, and cellular telephone technologies. The hypothesis that, at an implementation level, such error correction processes may be based on principles borrowed from the theory of non-linear dynamics is discussed, placing them within the more general context of genetic information processing; second, the existence of a strong mathematical ordering inside the genetic code complicates to some extent the framework for the explanation of the origin of the code (and consequentlyof the origin of life): if it is difficult to understand the origin of the code within a necessarily short evolutionary time, it is still more puzzling to comprehend how a high degree of ordering has been attained in such a short time if a clear biological advantage cannot be associated with this mathematical structure. It must be remarked that the few other mathematical approaches aimed at describing the organization of the genetic code point to similar paradoxical conclusions (see also Chapter 7, this volume).