Beyond Linear Additive and Hill Functions: A General Logistic Reformulation of Delay-Coupled Gene Regulatory Networks with Equilibrium Analysis, Hopf Bifurcation, and Lipschitz Stability

Abstract

Hill functions, dominant in gene regulatory network modeling, carry fundamental limitations: at non-integer cooperativity exponents, routine when fitting dose-response data, derivatives diverge at the origin, complex arithmetic corrupts ODE trajectories, and zero output at zero activation traps models in off-states. This paper employs logistic-based models that are globally $C^\infty$, real-valued, and strictly positive at zero concentration, resolving all three pathologies while preserving sigmoidal dynamics. Using the delay-coupled two-gene mutual-activation and self-repression network of Vinoth et al.\ as a concrete model, we analyze two reformulations: linear additive activation with logistic self-repression, and a fully sigmoidal form with both terms logistic. A closed-form matching relation $λ= n/θ$ follows from equating slopes at half-maximal points. Closed-form parameters of the weighted logistic formulation are derived by matching basal rates and local slopes to the Hill-linear hybrid model. The unique biologically feasible equilibrium is computed in each case; it is lower in the weighted logistic case, the reduction arising from saturation of the bounded activation term. In the delay-free case ($τ=0$), local asymptotic stability holds in both formulations since the Jacobian trace is strictly negative for all positive parameters; stability persists for $τ\in(0,τ_c)$ and is lost via Hopf bifurcation at the critical delay $τ_c$. Numerical solution of the full transcendental system locates $τ_c$, with higher-order bifurcations characterised numerically in each case. Replacing linear additive with weighted logistic activation substantially reduces both the global Lipschitz constant of the right-hand side and that of its Jacobian, permitting larger integration steps.