A Universal Attractor Topology Governs Vertebrate Sex Determination

We analyze bistable switches, developmental decision boundaries, and oscillatory manifolds within a unified dynamical framework based on attractors, ridges, and noise-shaped trajectories. Systems as diverse as the lambda switch, mammalian sex determination, the synthetic toggle, and the repressilator share a common geometric grammar: each organizes behavior around stable attractors separated by ridges whose curvature, depth, and asymmetry determine decision reliability, noise sensitivity, and entrainment flexibility. Using deterministic ODE models, global parameter sweeps, stochastic Langevin simulations, and ridge-based geometric analysis, we show that this kernel generates a mechanistic atlas spanning four dynamical regimes—a robust binary plateau, a narrow mosaic band, a dysgenesis wedge, and shifted regions corresponding to fate bias—that map directly onto biological and clinical phenotypes. For the sex-determination circuit, varying Hill cooperativity and auto-activation strength recovers clinical differences of sex development (DSDs), including SOX9 enhancer alterations, NR5A1 loss, WNT4/RSPO1 disruption, and FOXL2 hypomorphs. The model predicts a finite SRY timing window required for XY fate commitment and noise-driven basin asymmetry that biases late reversals toward the female fate under reduced Dmrt1. For oscillatory motifs, phase reduction yields a smooth infinitesimal phase response curve (iPRC) and a wide 1:1 Arnold tongue, establishing the repressilator as a robust timing module whose noise signature—phase diffusion along the limit cycle—is the oscillatory analogue of switching in bistable systems. Ridge curvature, iPRC shape, and noise-induced trajectory statistics provide consistent quantitative metrics for comparing these tradeoffs across motif classes, establishing a unified grammar that reveals deep homology across disparate systems and offers a principled basis for predicting how regulatory architecture shapes functional performance in natural and engineered contexts.