Ghost Dynamics in Receptor Signalling Networks: A Fast--Slow Adaptive Extension of Competitive Cancer Inhibition Models

Abstract

Receptor occupancy models quantify inhibition of cancer signalling, often treating target occupancy as a proxy for downstream activity. Although this simplification yields low-dimensional models, it cannot represent delayed pathway shutdown, transient resistance, or non-monotone viability responses generated by intracellular networks. We formulate a fast--slow framework that distinguishes drug--target occupancy from downstream signalling activity. The fast variable $X$ represents occupancy, the activity variable $A$ represents pro-survival signalling, and the slow variable $B$ represents adaptive feedback including phosphatase induction, stress adaptation, or signalling rewiring. Rapid occupancy relaxation permits a quasi-steady reduction, while weak feedback guarantees global convergence to a unique equilibrium. Stronger feedback may bring the frozen activity subsystem near a saddle-node fold, producing a ghost regime in which trajectories linger near the remnant of a vanished high-activity equilibrium. We determine when slow adaptive passage preserves the inverse-square-root ghost law or, under transverse crossing, produces the dynamic delay scale $O({\epsilon }^{-1/3})$. Coupling activity to viability translates these delays into shoulders in dose--response curves, while adaptive lag may generate early-time overshoot without ad hoc forcing. A proof-of-concept fit to time-resolved viability data reproduces patterns across measured concentrations and observation times and reveals practical identifiability limitations. The framework also predicts exposure-time-dependent shifts in apparent potency, including $I{C}_{50}(t)$, because viability integrates signalling activity over time rather than receptor occupancy alone.