Fig. 1.

Standard two-state promoter model (Model 1): activator fidelity is bounded by the Hopfield barrier. (A) Transition graph of Model 1. (B) Activator fidelity approaches its upper limit or Hopfield barrier, $f_0=e^{-\mathrm{\Delta \Delta }G°/RT}$, as the activator on rate, $\kappa$, tends to zero. To calculate the graph, we assumed $k_C=1$, $k_i=100$. Actual fidelities must be markedly lower than $f_0$: for instance, measured off-rates for Pho4 of yeast (the activator of PHO5) for specific and nonspecific sequences are ∼0.01 and 1 s⁻¹, respectively (7, 9). From Pho4’s equilibrium dissociation constant for correct binding of $K_d=11$ nM (9), and nuclear concentration of $\sim 60$ nM (47) (assuming a nuclear volume of 4 femtoliters), both the on-rate, $\kappa =0.06{\text{s}}^{-1}$ (indicated by a vertical line), and relative fidelity (indicated by horizontal line) may be calculated; the unit on the abscissa, then, is $0.01{\text{s}}^{-1}$. (C) Representative “sample path” (single cell trajectory of mRNA abundance) at relative activator fidelity of 0.95; the sample path was obtained with the Gillespie stochastic simulation algorithm (48) with $\kappa =0.05$, $k_C=1$, $\delta =0.1$ (rate constant for mRNA degradation), and average rate of transcription, $v_C=5$. (D) The Fano factor tends to infinity as activator fidelity, $f_1$, approaches the Hopfield barrier $\left(i.e.,f_1/f_0=1\right)$. Calculations were based on the assumption of $k_C=1$, $\delta =0.1$, and average rate of transcription $v_C=5$. Both Fano factor and fidelity were calculated as functions of the activator on-rate, κ (SI Appendix).