Molecular Machines as Origami ISA Programmes: Ribosomes, Photosynthesis, and Nitrogenase as Pachner-Move Computers — A Primer for Biologists and Biochemists

Molecular Machines as Origami ISA Programmes: Ribosomes, Photosynthesis, and Nitrogenase as Pachner-Move Computers

A primer for biologists and biochemists — no quantum mechanics or representation theory required.

Three of the most important molecular machines in biology — the ribosome, the FMO photosynthetic complex, and nitrogenase — share an unexpected mathematical structure: their active sites have the combinatorial geometry of the Fano plane, the smallest projective plane (7 points, 7 lines, 3 points per line). This paper shows that each machine is running a programme in the Origami ISA, a five-opcode instruction set whose opcodes are elementary geometric operations (Pachner moves) on a triangulated network. The same mathematical object — the 6j symbol of angular momentum theory — governs codon recognition in the ribosome, exciton transfer in FMO, and spectroscopic transitions from hydrogen to californium.

The ribosome is a 6j-symbol evaluator. The codon–anticodon recognition step at the A-site evaluates the 6j symbol on the network of 7 key residues (A1492, A1493, G530, A1913, C518, U531, and the codon base). Cognate (correct) tRNAs produce the trivial output; near-cognate tRNAs produce a non-trivial output that triggers rejection. A structural search over all 5040 permutations of the 7 residues onto the 7 Fano points identifies 6/7 Fano-line coverage in the cognate state (PDB: 4V9D, T. thermophilus 70S) versus 5/7 in the empty and near-cognate states. The broken line is the A1913–A1492 inter-subunit coupling spanning the 30S–50S junction — unique among the seven residue pairs. This is a parameter-free structural prediction.

FMO photosynthesis runs the uniquely efficient topology. The Fenna–Matthews–Olson complex achieves η ≈ 0.18 energy transfer efficiency with the BChl-3/BChl-4 coupling as the weakened line (coupling ratio r ≈ 0.18). A theorem (proved in Paper 325) states that among all connected 7-node graphs, the broken-Fano topology is the unique graph with positive information-geometric Carnot efficiency η > 0. With r = 0.18 from the crystal structure as the only input:

η = 1 − 𝒮_cold / 𝒮_hot = 0.1825

No free parameters. The experimentally measured value is η ≈ 0.18. Evolution found this topology independently in three molecular machines because it is the only solution to the problem of directed information flow in a 7-component system.

Nitrogenase breaks N≡N using G₂ triality. The [7Fe-9S-Mo] FeMo-cofactor has the exact combinatorial structure of the Fano plane (7 Fe atoms = 7 Fano points; 7 bridging sulfide interactions = 7 Fano lines). The N≡N bond-breaking step requires the SPIN opcode — the generator of triality, the order-3 outer automorphism of Spin(8) whose fixed-point subgroup is G₂ = Aut(𝕆). Standard quantum chemistry codes use associative matrix algebra (SU(2), SU(3)), which is provably insufficient for this non-associative transition. The topological defects of the G₂ gauge field — virtual monopoles at Fe4/Fe5 — lower the activation barrier by providing a topological pathway not available to associative electronic structure. Testable predictions: transient Fe4–Fe5 distance change by time-resolved EXAFS; asymmetric hyperfine splitting in ⁵⁷Fe Mössbauer spectroscopy during turnover.

Circuit sizes. Every biological cofactor of current spectroscopic interest can be simulated by an Origami ISA circuit of at most 21 qubits: single transition-metal ion (3 qubits), Fe₂S₂ cluster (7 qubits), FeMo-cofactor Fe₇Mo (21 qubits), FMO BChl₇ (21 qubits).

The paper is written as an entry point for biologists and biochemists. All quantitative results are referenced to primary papers (318, 324, 325, 347, 348, 374). The nitrogenase mechanism (§4) is the most speculative result; the ribosome and FMO results (§2–3) are supported by structural data and parameter-free quantitative predictions.