Figure 6.

Impact of the promoter dynamic range on the metabolic steady state and the modes of the feedback system. (a) Steady-state value of the metabolic flux, and the concentrations of the product and the first enzyme. (b) Location of the promoter-dependent eigenvalues λ_prom. The plots were generated by solving the steady-state equation (3.2) (a) and computing λ_prom from (3.13) (b) for a different dynamic range μ spanning five orders of magnitude and a fixed tightness κ⁰. The real and imaginary parts in (b) are normalized to the degradation rate γ. We considered a pathway with two metabolites and two enzymes with Michaelis–Menten kinetics (g_i(s_i−1) = k_{cat i} s_i−1/(K_{M i} + s_i−1)) with k_{cat i} = 32s⁻¹ and K_{M i} = 4.7 µM. These are representative values for PRA isomerase (extracted from the BRENDA database [29], EC number 5.3.1.24), a transcriptionally regulated enzyme in the tryptophan pathway of E. coli. We took the enzyme degradation rate as 2 × 10⁻⁴ s⁻¹ (half-life h), and used a substrate concentration s₀ = K_M1 (so that g₁(s₀) is at half-saturation). We modelled the product consumption as a Michaelis–Menten function d(s_n) = d_maxs_n/(K_d + s_n) with d_max = 24.96 µM min⁻¹ and K_d = 0.2 µM, both taken from an experimentally validated model for the tryptophan pathway [20]. Promoter tightness was fixed to κ⁰ = 0.03 nM min⁻¹ and the RBS strength to b₁ = 1, whereas the repression function was described by a Hill function σ(s_n) = θ^h/(θ^h + s_n^h) with Hill coefficient h = 2 and repression threshold θ = K_d. (Online version in colour.)