A Minimal Toggle-Switch Model of an Epigenetic Locking Parameter: Bistability Loss via a Saddle-Node Bifurcation, with an Application to Framing the Regenerative-Capacity Question in Adult Mammals

Adult mammals, unlike urodele amphibians, cannot regenerate complete limbs. A common heuristic explanation invokes a progressive "locking" of cellular identity during development. This paper develops a minimal, fully transparent, and independently reproducible dynamical-systems formalization of that heuristic. We model two mutually repressing regulatory genes with a single scalar parameter θ intended to represent the strength of accumulated epigenetic and structural constraints. We show analytically and numerically that increasing θ drives a genuine saddle-node (fold) bifurcation: below a critical value θc the system is tristable (two stable cell-fate attractors separated by a saddle), and above θc the saddle collides with one stable attractor and annihilates it, leaving a single remaining fate. For the baseline parameter set used here, θc = 0.3661, a value that converges to five significant figures across Newton-solver tolerances spanning 10−8 to 10−12.We quantify parameter uncertainty three independent ways — Monte Carlo propagation, Sobol indices (Saltelli sampling), and eFAST — using a continuation-based branch-tracking estimator that follows the two specific fixed points responsible for the bifurcation from θ = 0 under parameter perturbation. Under ±15% uncertainty on the six rate constants: θc = 0.360 ± 0.141 (95% CI [0.090, 0.590], N = 271 valid draws of 300); βA dominates total-order sensitivity (ST = 0.994), with roughly 38% of the variance in θc arising from parameter interactions rather than additive main effects; and βB's role is method-dependent (Sobol attributes it mainly to interaction, eFAST to a first-order-like effect of ≈ 0.20), a discrepancy we discuss rather than resolve prematurely. We prove analytically, via the Bendixson–Dulac criterion, that this system admits no periodic orbits for any θ studied, ruling out oscillatory or chaotic behavior rigorously. We validate a Bayesian MCMC estimation pipeline (four chains, Gelman–Rubin R̂ ≈ 1.0002, effective sample size ≈ 5600) on synthetic pseudo-data generated from a known ground-truth θc, recovering that value to within 12%, so that the pipeline is ready to apply to real data from the protocol proposed in Section 6 once such data exist. We also compare how well standard tabular classifiers (logistic regression, random forest, a small neural network) recover the bistable/monostable boundary from labeled samples alone (83–97% held-out accuracy) against the exact mechanistic criterion (100% by construction). All reported numbers are produced by the code in Appendix A.We explicitly do not claim that this two-gene toy model constitutes a validated mechanistic explanation of human regenerative failure, that it is quantitatively superior to machine-learning approaches, or that its dynamics are chaotic: a continuous-time two-dimensional autonomous system cannot exhibit chaos, by the Poincaré–Bendixson theorem. What we present is a small, internally consistent, falsifiable formal model, together with a concrete wet-lab protocol (Section 6) that could in principle test whether an analogous locking parameter operates in real cells.Keywords: genetic toggle switch, saddle-node bifurcation, epigenetic locking, bistability, regenerative biology, dynamical systems, falsifiability, reproducibility