Learning the Cellular Dynamics as a Port-Hamiltonian System:A Composite Multi-Clock GNN-Surrogate for Multi-Omics Circadian Cell Biology

We present a compartmental, multi-clock port-Hamiltonian model of cell dynamics learned by a graph neural network. The state pairs the measured abundance deviation of each molecular species with an oscillatory phase coordinate, derived only for species a per-clock rhythmicity gate certifies as periodic. The stored regulatory energy decomposes over five functional compartments (core clock, redox, energy, signalling, biosynthesis), so thermodynamic stability is verified compartment by compartment. Two mechanistically distinct clocks are included --- the 24 hour transcription-translation feedback loop and the 20-hour transcription-independent redox oscillator --- coupled through a zero-net-power signalling link, with the central-dogma gene-to-protein correspondence hard-wired and conserved moiety pools held as exact invariants. We evaluate the model on a real mouse-liver three-omic circadian dataset from public repositories and report a mixed verdict. The trained model is thermodynamically stable (no violations across three seeds), forecasts held out trajectory segments (root-mean-square error 0.324), and recovers withheld regulatory edges above chance (AUROC 0.94). Its central prediction --- that cross-omic phase lags equal arctan of clock frequency divided by molecular degradation rate --- matches the aggregate transcript-to protein lag (5.7 versus 4.9 hours) but not the gene-to-gene variation. The framework gives a falsifiable, thermodynamically grounded account of cell dynamics with explicit limits.