Hidden Block-Triangular Structure Freezes Relaxation Spectra of Gene Expression

摘要: We show that the generator of a general feedback-free gene expression network, comprising an arbitrary number of promoter states, mRNA, and protein, acquires a strictly block-triangular form when expanded in a falling factorial basis and ordered by a global index $L=k+l$. This hidden algebraic structure renders the entire relaxation spectrum exactly solvable: all eigenvalues are given by $\{\mu_i - \delta_m k - \delta_p l\}$, where $\mu_i$ are the eigenvalues of the promoter switching matrix, $\delta_m$ and $\delta_p$ are degradation rates, and $k,l\in\mathbb{N}_0$. The spectrum is completely independent of transcription and translation rates---a phenomenon we term spectral freezing. The block-triangular form further implies an algebraic skin effect: all eigenfunctions possess strictly finite support in the factorial basis, confining the transient dynamics to an exactly closed finite-dimensional subspace. Accidental degeneracies between blocks, protected by the triangular structure, give rise to non-diagonalizable Jordan blocks and exact algebraic relaxation $t e^{-\lambda t}$ when synthesis rates are asymmetric. The structure generalizes to arbitrary biochemical cascades satisfying a causal hierarchy condition. We propose direct experimental tests using synthetic biology to detect feedback through measurable spectral shifts.