Symmetry-constrained spectral fibres: reverse-complement group actions on double-stranded DNA and the fibres of circular molecular fingerprints

Abstract

A representation that depends on an object only through its local content is constant on the object's spectral fibre, the set of objects with the same multiset of local pieces. This paper treats two settings in which a symmetry enters. Double-stranded DNA. Reverse complementation generates a group G = {e, ρ} acting on sequence space. A strand-agnostic local encoder of order k+1 is G-invariant and is therefore blind to the double-stranded fibre DS(S, k), which contains the one-strand fibres of both strands and, in general, much more. This fibre is a sum of BEST determinants over the points of a lattice, the (−1)-eigenlattice of ρ on the circulations of a double cover, and counting it is #P-hard (results of the monograph Algebraic Genomics, doi:10.5281/zenodo.22943487). For bacteriophage φX174 at k = 10 the fibre has 31,925,753,246,212 members, between the single-strand count 8,610,708,632 and the 1.4·10^36 Eulerian circuits of the double cover, which a naive application of the BEST theorem would give. Circular fingerprints. The environments hashed by Morgan (ECFP) fingerprints are read up to the automorphisms of their centre because the neighbours of an atom are unordered. We prove that on substituted linear chains the radius-r Morgan partition is exactly the spectrum of (2r+1)-windows read up to reversal, with the substituent at distance r coarsened to its atom type, and verify this exhaustively with RDKit on all 19,683 chains CH3(CHR)9CH3, R in {H, F, Cl}, at radii 1–3. Among the 9,963 distinct molecules, 3.2% share their ECFP4 with another molecule and none share ECFP6. We construct distinct molecules with identical ECFP4 and ECFP6 (2048-bit, count and unhashed) by exchanging substituents across repeated (CH2)_l linkers; the fingerprints coincide iff l ≥ 2r−1, and the bound is attained. In the linker-4 pair the topological distance between the chlorine and the hydroxyl is 7 bonds in one molecule and 12 in the other. Stereoisomers are fibres of the default fingerprint that the chirality option removes; the linker pairs are not. A common quotient framework covers composition encoders, convolutional encoders, strand-agnostic DNA encoders and circular fingerprints. This record contains the paper (PDF and LaTeX source), the RDKit code and its transcript, and the φX174 genome (NC_001422.1). The φX174 counts are those of the verification scripts of the monograph's repository. Companion papers: doi:10.5281/zenodo.22950095, doi:10.5281/zenodo.22959211, doi:10.5281/zenodo.22960439, doi:10.5281/zenodo.22963882, doi:10.5281/zenodo.22966020. Code: https://github.com/Ruqing1963/symmetry-fibres