A Bounded Stochastic Extended Boolean Model for Signal Amplification, Noise Propagation, and Feedback Oscillation in Cell Signaling Pathways

Extended Boolean models provide a computationally efficient way to represent graded protein-protein interaction strengths in cell signaling pathways. Previous stochastic extended Boolean models based on the Waller-Kraft operator reproduced downstream signal amplification, stochastic noise propagation, and, after the inclusion of negative interactions, oscillatory and compensatory responses. However, these formulations were primarily simulation-based: protein activities were interpreted as normalized variables in [0,1], but the stochastic differential equations did not guarantee invariance of this biological state space, and the key amplification and feedback phenomena were not derived analytically. Here we introduce a bounded stochastic extended Boolean model in which activation and deactivation/inhibition inputs are separated and the noise is degenerate at both biological boundaries. We prove well-posedness, positive invariance of [0, 1]^N, and strict preservation of the open biological state space. We further derive closed moment equations for a low-activation linearization, obtain explicit downstream mean and coefficient-of-variation (standard deviation divided by a positive mean) amplification results for feed-forward cascades, and identify a linear Hopf-type stability threshold for a minimal negative-feedback motif. Motif-level numerical benchmarks verify the analytical results by comparing boundary preservation, moment equations versus Monte Carlo simulation, and the Hopf-type loop-gain threshold. Additional simulations assess approximation error and parameter sensitivity. An extended noise scan distinguishes comparatively stable mean suppression from increased ensemble relative variability at larger noise amplitudes. A Fucoxanthin-like canonical-module perturbation in a nuclear factor κB (NF-κB) network reduces p50/p65 and tumor necrosis factor α (TNFα) more than noncanonical readouts. This illustrates preferential attenuation of a modeled inflammatory branch with relative preservation of parallel outputs; magnitudes depend on rates, and branch preference reflects perturbation placement rather than validated drug selectivity.