Exploring Logistic Functions as Robust Alternatives to Hill Functions in Genetic Network Modeling

Abstract: Gene regulatory networks exhibit sigmoidal dynamics traditionally modeled using Hill functions. When Hill coefficients are non-integer values--ubiquitous in experimental fitting--these functions lose differentiability at low expression, creating singularities that compromise numerical stability and impede control applications. We present a systematic framework replacing Hill functions with logistic functions: increasing for activation, decreasing for repression. Logistic functions preserve sigmoidal characteristics while offering key advantages: infinite differentiability, closed-form derivatives simplifying Jacobians, invertible forms enabling feedback linearization, and built-in basal expression. We prove existence and uniqueness with explicit Lipschitz bounds, guaranteeing unique solutions and boundedness. Parameter estimation with biologically motivated thresholds demonstrated in case studies: genetic oscillators, positive autoregulation in E. coli, two-gene chaotic networks. Simulations with experimental parameters show: logistic models allow noise escape from low-expression traps via basal, while Hill models trap irreversibly--relevant to gal operon and bistables. Logistic functions respond to absolute concentrations rather than logarithmic fold changes, aligning with molecule count-based decisions. Control advantages: controllability at zero (missing in Hill), seamless MPC integration, superior stability. Logit extensions enable network inference from scRNA-seq data, using concavity for convergence and handling dropouts. Applications: immunology, hematopoiesis with delays, environmental systems. The framework advances modeling for synthetic biology, therapeutic interventions, metabolic engineering, and genome-scale analysis.