Logistic Gene Regulatory Networks: Prevention of Expression Shutdown, and Numerical Stability Beyond Hill Function

TL;DR

This paper introduces logistic functions as a superior alternative to Hill functions for modeling gene regulatory networks, addressing key structural flaws and enhancing numerical stability and biological realism.

Contribution

The authors propose logistic functions for gene regulation modeling, demonstrating their advantages over Hill functions in stability, biological interpretability, and computational robustness.

Findings

Logistic functions are globally smooth and real-valued, avoiding Hill function flaws.

Stability analysis shows no Hopf bifurcation without delays in logistic models.

Logistic models successfully simulate complex gene networks without numerical issues.

Abstract

Hill functions, the standard tool for modelling gene regulatory networks, carry three structural flaws when the cooperativity exponent is non-integer: loss of global smoothness, silent complex-valued arithmetic corruption of ODE trajectories, and an identically zero basal production rate that traps bistable models in off-states. Logistic functions $f^\pm$, being globally $C^\infty$, real-valued for all arguments, and strictly positive at zero, resolve all three simultaneously. For a two-gene negative-feedback oscillator, local asymptotic stability is established for all positive parameters via the Routh--Hurwitz criterion, and no Hopf bifurcation is possible without time delays. For bistable positive autoregulation, saddle-node thresholds are characterised through explicit transcendental equations; with biophysically grounded \textit{E.~coli} parameters, basal logistic production drives off-state escape in $\approx 44$~min while the Hill model remains permanently trapped. The 11-gene Traynard cell-cycle Boolean network is translated automatically via the product-of-logistics De~Morgan formalism and integrated without warnings, all variables remaining bounded and non-negative. The De~Morgan framework places every repressor threshold at a positive measurable concentration, whereas the weighted-sum formulation of Samuilik et al.\ places repressor critical points at negative concentrations, rendering them biologically inert. On an 80-gene Boolean-derived ODE system with $n = 3.509$, the Hill solver entered silent complex-valued contamination at $t \approx 52.64$ and terminated near $t \approx 63$--$65$; the logistic formulation completed $t \in [0, 200]$ without a single warning. The always-positive production rate ensures full controllability, enabling sliding mode, model predictive, and feedback-linearisation strategies where Hill-based formulations fail.