Logistic Gene Regulatory Networks: A Modelling Framework Beyond Hill Functions

Abstract

Boolean networks model gene regulatory networks, but extracting quantitative dynamics requires translating their logical rules into differential equations, and the sigmoidal kernel chosen carries direct biological consequences. The near-universal choice, the Hill function, sets production to exactly zero when an activator is absent, creating a spurious absorbing off-state with no biological counterpart. We develop a product-of-logistics framework in which increasing logistic functions represent activation, decreasing logistic functions represent repression, and a recursive De Morgan product formula translates an arbitrary Boolean rule into a continuous regulatory function. The translation is automatic, confines every regulatory function to the unit interval, and retains a strictly positive basal rate. Our central result is a recovery theorem: every steady state of the Boolean network reappears, for sufficiently steep response, as an exponentially stable equilibrium of the continuous model, so the translation provably refines rather than distorts the Boolean analysis. We establish global well-posedness, forward invariance, and an explicit Lipschitz constant, and prove, for the two canonical two-gene motifs, global asymptotic stability of the negative-feedback oscillator and a closed-form bistability threshold for the toggle switch. Every threshold remains a positive, measurable concentration, unlike weighted-sum logistic formulations that place repressor thresholds at meaningless negative values. The eleven-gene Traynard mammalian cell-cycle network is translated automatically: in the proliferative regime its trajectories settle onto a sustained limit cycle reproducing the Boolean cyclic attractor. Because it is purely structural, the translation applies unchanged to existing Boolean models and supports exact feedback linearisation for control.