Figure 1. Strategy for measuring TF-DNA interactions.

(A) A thermodynamic model of simple repression. Here, promoter DNA can transition between three possible states: unbound, bound by a TF, or bound by RNAP. Each state has an associated Boltzmann weight and rate of transcript initiation. $F$ is the TF binding factor and $P$ is the RNAP binding factor; see text for a description of how these dimensionless binding factors relate to binding affinity and binding energy. $t_{\mathrm{sat}}$ is the rate of specific transcript initiation from a promoter fully occupied by RNAP. (B) Transcription is measured in the presence ($t_{+}$) and absence ($t_{-}$) of the TF. Measurements are made for an allelic series of RNAP binding sites that differ in their binding strengths (blue-yellow gradient). (C) If the model in panel A is correct, plotting $t_{+}$ vs. $t_{-}$ for the promoters in panel B (colored dots) will trace out a 1D allelic manifold. Mathematically, this manifold reflects Equation 1 and Equation 2 computed over all possible values of the RNAP binding factor $P$ while the other parameters ($F$, $t_{\mathrm{sat}}$) are held fixed. Note that these equations include a background transcription term $t_{\mathrm{bg}}$; it is assumed throughout that $t_{\mathrm{bg}}\ll t_{\mathrm{sat}}$ and that $t_{\mathrm{bg}}$ is independent of RNAP binding site sequence. The resulting manifold exhibits five distinct regimes (circled numbers), corresponding to different ranges for the value of $P$ that allow the mathematical expressions in Equations 1 and 2 to be approximated by simplified expressions. In regime 3, for instance, $t_{+}\approx t_{-}/(1+F)$, and thus the manifold approximately follows a line parallel (on a log-log plot) to the diagonal but offset below it by a factor of $1+F$ (dashed line). Data points in this regime can therefore be used to determine the value of $F$. (D) The five regimes of the allelic manifold, including approximate expressions for $t_{+}$ and $t_{-}$ in each regime, as well as the range of validity for $P$.