Evolution of the genetic code through progressive symmetry breaking

Abstract

Evolution of the genetic code in an early RNA world is dependent on the steadily improving specificity of the coevolving protein synthesis machinery for codons, anticodons, tRNAs and amino acids. In the beginning, there is RNA but the machinery does not distinguish yet between the codons, which therefore all encode the same information. Synonymous codons are equivalent under a symmetry group that exchanges (permutes) the codons without affecting the code. The initial group changes any codon into any other by permuting the order of the bases in the triplet as well as by replacing the four RNA bases with each other at every codon position. This group preserves the differences between codons, known as Hamming distances, with a 1-distance corresponding to a single point mutation. Stepwise breaking of the group into subgroups divides the 64 codons into progressively smaller subsets - blocks of equivalent codons under the smaller symmetry groups, with each block able to encode a different message. This formalism prescribes how the evolving machinery increasingly differentiates between codons. The model indicates that primitive ribosomes first identified a unique mRNA reading frame to break the group permuting the order of the bases and subsequently enforced increasingly stringent codon-anticodon basepairing rules to break the subgroups permuting the four bases at each codon position. The modern basepairing rules evolve in five steps and at each step the number of codon blocks doubles. The fourth step generates 16 codon blocks corresponding with the 16 family boxes of the standard code and the last step splits these boxes into 32 blocks of commonly two, but rarely one or three, synonymous codons. The evolving codes transmit at most one message per codon block and as the number of messages increases so does the specificity of the code and of protein synthesis. The selective advantage conferred by better functioning proteins drives the symmetry breaking process. Over time paralogous tRNA evolution expands the anticodon repertoire, which is divided into anticodon blocks matching the codon blocks under the stage-specific ribosomal basepairing rules. Contemporaneously an expanding family of primitive aminoacyl-tRNA synthetases (aaRSs) divides the tRNA diversities into various different and overlapping subsets: each aaRS accepts some tRNAs but rejects all others and several aaRSs may accept the same tRNA species. Selection favoring less ambiguous codes eliminates these overlaps and also imposes the ribosomal anticodon block division as ambiguity arises when different aaRSs accept tRNAs of the same anticodon block. Only when the tRNAs of one or several anticodon blocks are accepted by a unique aaRS does the code become specific. This coding pattern is observed in the standard code and the evolution of amino acid assignments by primitive aaRSs onto tRNAs is traced back via tRNA trees that picture a gradual division of tRNA diversities into blocks with increasingly specific amino acid assignments. Symmetry breaking combined with...

The Graph, Geometry and Symmetries of the Genetic Code with Hamming Metric

“…At each stage of evolution the symmetries of the faces listed, which are the stabilizer groups of the vertex sets spanning them, are exact code symmetries: the vertices of a face and the codons they represent are equivalent under the stabilizer group, these codons encode the same message. The splitting of the CodonPolytope into smaller faces corresponds with the breaking of the symmetry group of the polytope into the smaller stabilizer groups of the faces, and the polytope splitting model is fully compatible with, and illustrates a more mathematically abstract symmetry breaking model for the evolution of the code [24]. Both models progressively partition the codon set in binary fashion; the five steps above split the polytope into smaller faces containing fewer vertices: [64] Non-saltatory, gradual evolution proceeds in small steps, and in a pre-LUCA, primitive RNA world, the emergent protein synthesis apparatus needs to evolve capacities to recognize codons, anticodons, tRNAs and amino acids.…”

“…Two vertices incident on the same edge are adjacent. The CodonGraph is comprised of 64 vertices, representing the codons, and 288 edges between adjacent vertices [24]. Adjacent vertices represent codons at Hamming 1-distance, and the graph is 9-regular as each vertex is adjacent to nine vertices representing the nine nearest neighbor codons at 1-distance.…”

“…Distance-0 entries (black) fall on the main diagonal, distance-1 (dark gray) entries correspond with edges between vertices i and j of the CodonGraph ( Figure 5). Each row/column contains one 0-distance; nine 1-distances; 27, 2-distances; and 27, 3-distances [24]. …”

“…The symmetry group of the graph thus equals S3 xwreath (S4)i × (S4)j × (S4)k of order 82,944. (The CodonGraph symmetry group was derived in a different way in [24].) The graph symmetry group contains stabilizer subgroups for all its 3376 subgraphs; they are isomorphic to the stabilizer groups for the polytope faces, and easily derived from the subarrays, as discussed in Section 4.4.…”

“…We use [n] as notation for a finite n-set = {1, 2, …, n} so that the codons can be indexed (numbered) by [64] = {1, …, 64} and the messages by [21]. The genetic code is one of 1.51 × 1084 different [64] → [21] surjections, an astronomically large function space [24]. (Appendix A summarizes the notation and combinatorial counting formulas used in this article, see [25]) The fact that just one code (or at most a few, very similar codes) evolved is a strong argument in favor of a last unique common ancestor (LUCA) for all known living organisms, but how, pre-LUCA, this unique code evolved is a yet unanswered question.…”