Influence of non-Gaussian noise on the coherent feed-forward loop with time delay

The effect of non-Gaussian noise on the coherent feed-forward loop with time delay is researched. Based on delayed Fokker–Planck equation, the expressions of the steady-state probability distribution, mean, normalized variance, normalized skewness, and normalized kurtosis of concentration variables y and z of two transcription factors X, Y and a target gene Z in the model are derived. Then, the impacts of the noise intensity D, parameter q indicating the departure from Gaussian noise, correlation time τ₀ of non-Gaussian noise, and time delay τ on the steady-state probability distribution, mean, normalized variance, skewness, and kurtosis are discussed. The results show that, non-Gaussian noise will weaken the stability of Y and Z, increase the concentration of y and z, and ultimately benefit the output of target gene Z. Time delay also weaken the stability of Y and Z, but increase the concentrations of y and decrease the concentrations of z, i.e., suppress the output of target gene Z. In addition, numerical simulations are performed and consistent with approximate theoretical results.