Fig. 2.

Theoretical analysis of fluctuations. a Schematic illustration of the two-stage stochastic gene expression model. v = production rate, d = degradation rate. (b) Modelling of three stochastic realizations of the KikGR reporter. After a 405 nm pulse the green fluorescence-emitting KikG is transformed into red fluorescence-emitting KikR. The auto-correlation between KikG at 3 h and 6 h is calculated. Parameters are: ν₀ = 2.25 h⁻¹, d₀ = 1.125 h⁻¹, d ₁ = 0.09 h⁻¹ and ν ₁ = 41.825 h⁻¹, ν ₁ = 48.506 h⁻¹, ν ₁ = 46.069 h⁻¹ for the three different trajectories. (c) Modelling of the non-stationary auto-correlation of the two-stage gene expression model in presence of extrinsic noise (crosses and triangles) as calculated from stochastic simulation of the KikGR reporter from 10⁵ trajectories (Supplementary Note 1) and the theoretical non-stationary auto-correlation of a birth-death process c₀(t ₂, t ₁) (black solid line) as a lower bound of the non-stationary auto-correlation (Supplementary Note 1). The extrinsic noise is simulated as cell-to-cell variations in the protein translation rate ν ₁ with different coefficients of variation (CV). Parameters for the two-stage model are: ν ₀ = 2.25 h⁻¹, d ₀ = 1.125 h⁻¹, d ₁ = 0.09 h⁻¹ and <ν ₁ > = 45 h⁻¹. In the case of no extrinsic noise (Var(ν ₁) = 0 h⁻², blue  triangles), the auto-correlation of the two-stage model approaches that of the birth-death model. With increasing extrinsic noise (Var(ν ₁) = 100 h⁻², red crosses) the auto-correlation increases. The reason for this is that the covariance and the variance become dominated by the extrinsic noise, for which a much longer correlation time was assumed