Seeded Assignments on the Chrysene Registry: Transversality, Third-Position Refinement, and the Mitochondrial Code as Normal Form

Abstract

A frame class is not a codon block: the two partitions of the sixty-four words are transverse, meeting in at most one word, so a residue attached to a class is not thereby attached to a codon. And by the parity obstruction of the preceding paper no map constant on classes can produce the standard genetic code, every class having size two or four while the code assigns methionine and tryptophan to one codon each. We exhibit a construction that crosses the gap: a seed map placing a class's residue at one codon, propagation filling the block, and a refinement dividing a block at the third base into its pyrimidine and purine halves.

Seventeen seeds suffice, with deficiency zero and three classes to spare; exactly eight hundred and forty seed maps achieve it and none is distinguished. The bound of twenty survives the crossing — every residue placed is seeded by a class — while the fibres are freed, so the parity obstruction does not apply. The construction outputs the vertebrate mitochondrial code, at every one of the sixty-four codons; the standard code is that output with two further divisions, at AUA and UGA. Its four stop codons are absences, two purine halves failing to liberate.

The central results concern what such a construction can produce. A block admits normal form — uniform, or two uniform halves — only if its residue pattern is (4) or (2,2), and the construction produces exactly those; normal form is therefore its range and not a regularity discovered of the genetic code. Of nine candidate refinements, three positions by three partitions, exactly one brings either code near normal form: refinement at the first or second position leaves all sixteen groups outside, under every partition. Holding each block's residues fixed and randomising within it, both codes attain the minimum their composition permits, at probabilities 1.4 x 10^-3 and 1.5 x 10^-4; arrangement is perfect in both and composition is where they differ. A hypothesis of the preceding paper is withdrawn, every stop codon of the output being an absence.

The test is reported at its score. The construction's own output is one of the four of twenty-seven published translation tables that lie inside the range, all four mitochondrial; twenty-three do not, and the further division they would require is needed fifty-one times at eight distinct blocks, of which this paper accounts for two. The parity obstruction is also bounded: it excludes the standard code and does not exclude the mitochondrial one, which has no odd fibre yet splits nineteen of twenty classes. Unlike its predecessor this paper uses the standard genetic code constructively, and says so.