Figure 4
Complexity of the steady-state. (A) This prokaryotic-like example corresponds to an energy-independent promoter regulated by two TF molecules (A and C) and the looping of DNA with typical parameters from the literature. A binds cooperatively at its two binding sites (ΔGA-A= -2 kcal/mol) and competitively with C at one of its sites (ΔGA-C= 1.5 kcal/mol) [68, 79]. The energetic cost of DNA looping (typically between 8 and 10 kcal/mol) is ΔGloop = 9 kcal/mol and is overcompensated by the interaction energy with two TFs of type A that maintain the loop (ΔGloop-A= -5.5 kcal/mol for each site) [67, 93, 94]. The closed state of DNA looping slows down the association/dissociation of C (ΔEloop-C= 2.5 kcal/mol). Bimolecular TF-DNA residence times were taken in the shorter range reported by [90] (1/
= 20 s at both sites and 1/
= 60 s) and the time for DNA to loop when both sites of A are occupied is very fast 1/kclose = 1 s [93]. Concentration ranges ([10-2; 103] nM) and equilibrium constants (
= 20 nM and 
= 1 nM) were set to physiological values [79, 89]. Transcription is promoted by the unlooped state, the presence of C and slightly by the presence of A at one site (see table S3 of Additional file 1 for details). RNA life-time (5 min) and abundance (between 10 and 70 copies per cell) were chosen as reported by [84, 91]. (B) Exploration of the system's behavior as a function of concentrations [A] and [C] is presented in terms of mean RNA level (B1), normalized variance (B2) and distribution (C) (represented along an arbitrary path of interest because of a too large dimensionality). (D) Changing the energies of activation E0 by adding a normally distributed energy (s.d. = 3 kcal/mol) to both direction of each reaction while keeping state energies G0 unchanged does not influence the mean behavior of expression but has a profound impact on its variability. This shows that mean expression can hide most of the complexity of regulation and that stochastic aspects can reveal much kinetic information.