Figure 2.

Theoretical (A–C) and experimental (D–F) results for the looping probability as a function of operator strength, loop sequence and repressor concentration. In the theoretical predictions of (A)-(C), K_id = 5.4 pM, K₁ = 16 pM and J_loop = 54 pM unless otherwise indicated; curves with these default parameters are shown as dashed blue lines for comparison across panels. In the experimental results of (D)-(F), unlike in (A)-(C), curves are fits to Equation 1, not predictions. (A) Theoretical prediction for the effect of changing the strength of one of the operators on the looping probability as a function of repressor concentration. (B) Theoretical prediction for the effect of changing the flexibility of the DNA in the loop. (C) Extension of the simple model to the case of two experimentally distinguishable looped states (see ‘Materials and Methods’ section and the section on loop length), which we model as having different J-factors. The two looped states are labeled ‘middle’ (‘M’) and ‘bottom’ (‘B’) in reference to their relative tether lengths. The dashed blue line shows the sum of the probabilities of the two states, which we refer to as the total looping probability. (D) Measured looping probabilities for 94 bp of the random E8 sequence, flanked by three different combinations of operators. Dashed lines indicate individual fits to each data set as described in Section S4.4 in the Supplementary Data; solid lines indicate a global fit to all the three data sets simultaneously. The global fit, which enforces identical values of the J-factor and O₁ dissociation constant in all the three data sets, describes the data as well as the individual fits, demonstrating the consistency of the model when the operators are changed. (E) Looping probabilities for the E8 (black) and TA (red) sequences as function of concentration. The Oid-E894-O1 data are the same as in (D); the dotted black line is the result of the global fit shown in that panel as well. The dashed red line represents an individual fit to the Oid-TA94-O1 data; the solid red and black lines are from a global fit to all three E8 data sets in (D) plus this TA data. (The results of this global fit that includes the TA data for the O1-E894-O1 and O2-E894-O1 data sets are shown in Section S4.4 in the Supplementary Data.) The TA data can be fit with the same K_d values as the E8 data, but have a significantly larger J-factor, or a more flexible sequence. Fit parameters for (D) and (E) are listed in Table 1. (F) Looping probabilities for a DNA with two looped states, Oid-E8107-O1. Curves represent a simultaneous fit of the ‘B’ and ‘M’ data to Equations S6 and S7, using the values of K_id and K₁ from the global fit to all three E8 data sets in (D) and the TA data in (E). The procedure for determining the errors on the fit follows the bootstrapping scheme used throughout this work and is described in Section S4.4 in the Supplementary Data. We find that the two looped states differ only in J-factor, as we and others (19,20) assume in our models; that is, the binding affinity of the repressor for operator DNA does not change with the different loop and/or repressor conformations that generate the two observed loop states. Note that the total J-factor of 330 pM obtained from this concentration curve is within error of the J-factor of 280 ± 40 pM determined from only the 100 pM data point shown in Figure 3C; likewise, the J-factors for the two looped states are within error of those determined from the 100 pM data alone (Supplementary Figure S8B).