Abstract
The Kummer surface was constructed in 1864. It corresponds to the desingularization of the quotient of a 4-torus by 16 complex double points. Kummer surface is known to play a role in some models of quantum gravity. Following our recent model of the DNA genetic code based on the irreducible characters of the finite group G5:=(240,105)≅Z5⋊2O (with 2O the binary octahedral group), we now find that groups G6:=(288,69)≅Z6⋊2O and G7:=(336,118)≅Z7⋊2O can be used as models of the symmetries in hexamer and heptamer proteins playing a vital role for some biological functions. Groups G6 and G7 are found to involve the Kummer surface in the structure of their character table. An analogy between quantum gravity and DNA/RNA packings is suggested.
Highlights
In a recent paper we found that the 22 irreducible characters of the group G5 :=
Let us illustrate our description with the genus 1 hyperelliptic curve introduced in the context of our model of the genetic code based on the group G5 := (240, 105) in which h(x) = 0 and f (x) = x4 − x3 − 4x2 + 4x + 1 [Section 5] of [1]
Since the Kummer surface appears in our models of DNA/RNA packings of some protein complexes such as the hexamer and pentamer rings one can ask the question whether quantum gravity is relevant in such biological realms
Summary
In a recent paper we found that the 22 irreducible characters of the group G5 :=. (240, 105) ∼= Z5 2O, with 2O the binary octahedral group, could be made in one-to-one correspondence with the DNA multiplets encoding the proteinogenic amino acids [1]. There exists a ubiquitous family of RNA-binding proteins called LSM proteins whose function is to serve as scaffolds for RNA oligonucleotides, assisting the RNA to maintain the proper three-dimensional structure Such proteins organize as rings of six or seven subunits. It is known that in the process of transcription of DNA to proteins through messenger RNA sequences (mRNAs), there is an important step performed in the spliceosome [5] It consists of removing the non-coding intron sequences for obtaining the exons that code for the proteinogenic amino acids.
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