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

TL;DR

This paper introduces logistic-based models for gene regulatory networks that overcome limitations of Hill functions, providing globally smooth, positive, sigmoidal dynamics with equilibrium and bifurcation analysis.

Contribution

It develops a fully sigmoidal reformulation of delay-coupled gene networks, derives closed-form parameters, and analyzes stability and bifurcations, improving modeling robustness.

Findings

Logistic models resolve Hill function pathologies.

Equilibrium is lower in weighted logistic models due to saturation.

Stability persists under delays until a critical Hopf bifurcation.

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 $\lambda = n/\theta$ 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 ($\tau=0$), local asymptotic stability holds in both formulations since the Jacobian trace is strictly negative for all positive parameters; stability persists for $\tau\in(0,\tau_c)$ and is lost via Hopf bifurcation at the critical delay $\tau_c$. Numerical solution of the full transcendental system locates $\tau_c$, 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.