<p>We have presented a unified theoretical framework for the turnover of stochastic kinetics and non-equilibrium thermodynamics of phosphorylation–dephosphorylation cycles (PdPc) and their extension to cascade signaling networks. Starting from the chemical master equation, we have employed a large deviation ansatz to recast the dynamics into the Hamilton–Jacobi formalism, yielding an explicit Hamiltonian with species concentrations and their conjugate momenta as canonical variables. This formulation provides a systematic evaluation of reaction velocities, fluctuation-induced diffusion, and entropy production rates (EPR). Our results reveal hallmark features of ultrasensitivity: in PdPc, sharp switch-like transitions are reflected in the slope of the EPR, which steepens with increasing chemical potential difference. In a typical PdPc cascade circuits within mitogen-activated protein kinase (MAPK), cooperative enzymatic interactions amplify these effects, optimized at a critical size of the cycle number.</p> Graphical abstract <p>An unidirectional signaling cascade is illustrated where receptor activation triggers sequential downstream cycles. Time-series and steady-state analyses show a sharp transition beyond the third cycle, highlighting cycle three as optimal. Increasing cycle numbers enhance cooperativity, sharpening transitions in system velocity and entropy production, indicating critical behavior in the cascade dynamics.</p> <p></p>