Learning the Cellular Dynamics as a Port-Hamiltonian System

We present a composite, compartmental, multi-clock port-Hamiltonian model of cell dynamics learned by a graph-neural-network surrogate. The state pairs abundance deviation qj, the quantity omics assays measure, with oscillatory phasors derived only for pools a rhythmicity gate certifies as periodic. The storage function decomposes over five functional compartments (core clock, redox, energy, signalling, biosynthesis), so passivity is certified compartment by compartment, and the model carries two mechanistically distinct clocks coupled through a zero-net-power signalling port, with the central dogma hard-wired and conserved moieties held as exact invariants. We evaluate it on a real mouse-liver tri-omic circadian dataset assembled from public repositories and report a deliberately mixed verdict. The trained model is passive ([Formula], no violations over three seeds), forecasts held-out trajectory segments (RMSE 0.324 {+/-} 0.0004), and recovers withheld regulatory edges above a permuted null (AUROC 0.94 {+/-} 0.01). Its central prediction -- that cross-omic phase lags follow {Delta}{varphi} = arctan({omega}/kdeg) -- matches the aggregate transcript-to-protein lag measured independently (5.7 vs 4.9 h) but not the gene-to-gene variation, and the internal two-clock cascade is not scoreable on the available cross-cohort metabolome. The framework thus gives a falsifiable, thermodynamically-grounded account of cell dynamics with explicit limits.