Linear noise approximation of noise-induced oscillation in NF-{kappa}B signaling network

NF-{\kappa}B, one of key regulators of inflammation, apoptosis, and differentiation, was found to have noisy oscillatory shuttling between the nucleus and the cytoplasm in single cells when cells are stimulated by cytokine TNF{\alpha}. We present the analytical analysis which uncovers the underlying physical mechanisms of this spectacular noise-induced transition in biological networks. Starting with the master equation describing both signaling and transcription events in NF-{\kappa}B signaling network, we derived the macroscopic and the Fokker-Planck equations by using van Kampen’s sysem size expansion. Using the noise-induced oscillatory signatures present in the power spectrum, we constructed the two-dimensional phase diagram where the noise-induced oscillation emerges in the dynamically stable parameter space.

💡 Research Summary

This paper investigates the phenomenon of noise‑induced oscillations (NIO) in the NF‑κB signaling network, a key regulator of inflammation, apoptosis, and cell differentiation. Experimental observations have shown that, upon stimulation with the cytokine tumor necrosis factor‑α (TNFα), single cells display stochastic shuttling of NF‑κB between the nucleus and the cytoplasm, producing apparent oscillatory behavior that cannot be explained by deterministic models alone.

The authors begin by constructing a comprehensive stochastic description of the NF‑κB pathway. All relevant molecular species—cytoplasmic NF‑κB, nuclear NF‑κB, IκB, A20, and the TNFα receptor complex—are represented as discrete copy numbers. Each biochemical reaction (binding, phosphorylation, degradation, transcription, translation, and transport) is assigned a propensity function, yielding a master equation that governs the time evolution of the joint probability distribution of the system state.

To extract analytical insight, the van Kampen system‑size expansion is applied. By decomposing the copy‑number vector (X) into a deterministic concentration component (\phi) (order (\Omega), where (\Omega) denotes cell volume) and a stochastic fluctuation component (\xi) (order (\Omega^{1/2})), the authors separate the dynamics into two hierarchical levels. The zeroth‑order term recovers the conventional deterministic rate equations, which describe the average behavior of the network. The first‑order term yields a linear Langevin equation for the fluctuations:

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