Figure 5.

Phage λ cI self-regulation kinetics. (A) The phage λ cI–DNA assembly network can we viewed as a module within the network that controls the production of cI repressors at the P_rm promoter. Binding of cI dimers (cI₂) to r3 (s_r3=1), represses transcription, whereas binding to r2 when r3 is unoccupied (s_r2=1 and s_r3=0) activates transcription. The transcription rate is given by k_t=(k_actnl(1−s_L)s_r2+k_actls_Ls_r2+k_bas)(1−s_r3), with k_actnl=0.05 s⁻¹, k_actl=0.0225 s⁻¹, and k_bas=0.005 s⁻¹; cI mRNA is degraded and translated into cI protein at rates k_mdeg=0.005 s⁻¹ and k_p=0.05 s⁻¹ per mRNA, respectively; and cI monomers and dimers are degraded at a rate k_deg. The cI dimerization reaction takes place with association and dissociation rate constants k_a=7 × 10⁸M⁻¹s⁻¹ and k_dis=9.9 s⁻¹, respectively. cI dimers in solution can bind nonspecifically to DNA, with an equilibrium binding constant K_ns=2.0 × 10⁴ M⁻¹, or bind to one of the free operators with an association rate k_on=a[N], with a=7 × 10⁸M⁻¹s⁻¹. Note that the values used for the association rates are the typical ones of diffusion-limited reactions. The dissociation rates k_off depend on the free energy difference between the complexes with (s) and without (s′) the cI dimer via the detailed balance principle: k_off=k_one^{−(ΔG(s′)−ΔG(s))/RT}. As cellular volume, we have taken 10⁻¹⁵ l. The rates of looping and unlooping DNA are k_on^L=30 s⁻¹ (Vilar and Leibler, 2003) and k_off^L, respectively, with k_off^L obtained also from the detailed balance principle. (B) Time behavior of the number of cI dimers for the case with WT operators (red) and with no left operator (green) when (C) the cI degradation rate k_deg (in units of min⁻¹) is switched from 0.025 min⁻¹ to 0.4 min⁻¹. Time on the horizontal axis is given in hours. (D) Activity of the promoter P_r controlling a reporter gene. In this case, there is transcription at a rate k_t^rep=0.12(1−s_r1)(1−s_r2)s⁻¹ only when both r1 and r2 are free. The reporter mRNA translation, mRNA degradation, and protein degradation rates are k_p^rep=0.01 s⁻¹, k_mdeg^rep=0.005 s⁻¹, and k_deg^rep=0.005 s⁻¹, respectively.