Figure 4. Strategy for measuring TF-RNAP interactions.

(A) A thermodynamic model of simple activation. Here, promoter DNA can transition between four different states: unbound, bound by the TF, bound by RNAP, or doubly bound. As in Figure 1, $F$ is the TF binding factor, $P$ is the RNAP binding factor, and $t_{\mathrm{sat}}$ is the rate of transcript initiation from an RNAP-saturated promoter. The cooperativity factor $\alpha$ quantifies the strength of the interaction between DNA-bound TF and RNAP molecules; see text for more information on this quantity. (B) As in Figure 1, expression is measured in the presence ($t_{+}$) and absence ($t_{-}$) of the TF for promoters that have an allelic series of RNAP binding sites (blue-yellow gradient). (C) If the model in panel A is correct, plotting $t_{+}$ vs. $t_{-}$ (colored dots) will reveal a 1D allelic manifold that corresponds to Equation 4 (for $t_{+}$) and Equation 2 (for $t_{-}$) evaluated over all possible values of $P$. Circled numbers indicate the five regimes of this manifold. In regime 3, $t_{+}\approx {\alpha }^{\prime }{t}_{-}$ where ${\alpha }^{\prime }$ is the renormalized cooperativity factor given in Equation 5; data in this regime can thus be used to measure ${\alpha }^{\prime }$. Separate measurements of $F$, using the strategy in Figure 1, then allow one to compute $\alpha$ from knowledge of ${\alpha }^{\prime }$. (D) The five regimes of the allelic manifold in panel C. Note that these regimes differ from those in Figure 1D.