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The transient response to a stimulus and subsequent recovery to a steadystate are the fundamental characteristics of a living organism. Here we studythe relaxation kinetics of autoregulatory gene networks based on the chemicalmaster equation model of single-cell stochastic gene expression with nonlinearfeedback regulation. We report a novel relation between the rate of relaxation,characterized by the spectral gap of the Markov model, and the feedback sign ofthe underlying gene circuit. When a network has no feedback, the relaxationrate is exactly the decaying rate of the protein. We further show that positivefeedback always slows down the relaxation kinetics while negative feedbackalways speeds it up. Numerical simulations demonstrate that this relationprovides a possible method to infer the feedback topology of autoregulatorygene networks by using time-series data of gene expression.

Relaxation rates of gene expression kinetics reveal the feedback signs of autoregulatory gene networks