Logistic Gene Regulatory Networks: A Modeling Framework Beyond Hill Functions

Boolean network models are a widely used framework for describing gene regulatory networks across many biological systems, from the mammalian cell cycle to cancer-signaling and developmental decision circuits. A Boolean model already identifies the attractors of a network and their basins; what it cannot supply are the graded expression levels, transition timing, parameter sensitivities, and responses to continuously varying inputs that a continuous description adds. Obtaining these quantitative refinements requires translating the logical update rules into a system of ordinary differential equations, and the sigmoidal kernel chosen for that translation is a modeling decision with direct biological consequences. The near-universal choice, the Hill function, sets production to exactly zero when an activator is absent; yet genes are never fully silent, so this idealization introduces a spurious absorbing off-state with no biological counterpart. We develop a general 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-conjunctions, disjunctions, and negations-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 regulatory response, as an exponentially stable equilibrium of the continuous model, with the discrete labels 0 and 1 realized as basal and saturated concentrations, so the translation provably refines, rather than distorts, the original Boolean analysis. We establish the analytical foundations that the framework requires-global well-posedness, forward invariance, an explicit Lipschitz constant, and, for the two canonical two-gene motifs, both the global asymptotic stability of the negative-feedback oscillator and a closed-form bistability threshold for the genetic toggle switch-and we show that every regulator threshold remains a positive, experimentally measurable concentration, unlike weighted-sum logistic formulations that place repressor thresholds at biologically meaningless negative values. The eleven-gene Traynard mammalian cell-cycle network is translated automatically and integrated: in the proliferative regime, its trajectories settle onto a sustained limit cycle that reproduces the cyclic attractor of the underlying Boolean model. A second and larger curated network-a twenty-five-node geroconversion model in its type-2-diabetes variant-is translated by the same automatic procedure and converges instead to a stable equilibrium that coincides with a Boolean fixed point, demonstrating the recovery theorem on a real fixed-point attractor; removing the single feedback edge that the source model identifies as diabetes-specific recovers a second, disjoint fixed point of the same network, the proliferative phenotype, showing that the translation is sensitive to minimal, mechanistically motivated edits at the level of a single regulatory edge. Because the translation is purely structural, the same procedure applies without modification to existing Boolean models of cancer signaling and developmental transitions, and the framework further supports exact feedback linearization for control design.