Fig. 1.

Linear mapping approximation (LMA) and its application to steady-state conditions. a Illustration of the main idea behind the LMA namely to approximate the reversible (nonlinear) reaction between protein and promoter in the nonlinear GRN by a first-order (linear) reaction with an effective reaction rate in a linear GRN. b The upper figure shows a heatmap of Hellinger distance (HD) between the LMA and the exact probability distribution of protein numbers in steady-state conditions with parameters ρ_u = 10, σ_u = 0.01 for the nonlinear GRN shown in a. The exact distribution is reported in⁷. The bottom figure shows a heatmap of Λ, which is the ratio of the values of the two eigenvalues of the Jacobian of the deterministic rate equations of the nonlinear GRN in steady-state conditions. The red broken line denotes the contour line of Λ = 1. Note that the value of the HD is very small over a wide range of the ratio of time scales Λ indicating that the LMA’s accuracy is independent of time-scale separation. c A comparison of the LMA and exact steady-state distributions for Points A and Point B, marked as white stars, on the heatmap in b; note that Point A corresponds to the parameter set with the largest HD (ρ_b = 100, σ_b = 0.0126 with HD of 0.0478). Point B corresponds to ρ_b = 35, σ_b = 0.004 with HD of 0.0032