The quantitation of the stability of a protein-mediated “looped complex” of the Lac repressor and DNA containing two protein-binding sites whose centers of symmetry are separated by 11 helical turns (114 bp) was accomplished by footprint and gel mobility-shift titration techniques. Lac repressor binding to this DNA was only moderately cooperative; a cooperative free energy of -1.0 kcal/mol was calculated in a model-independent fashion from the individual-site loading energies obtained from the footprint titration studies. In order to partition the cooperative binding energy into components representing the dimer-tetramer association of Lac repressor and the cyclization probability of the intervening DNA, advantage was taken of the presence of experimental measures that were in proportion to the concentration of the looped complex present in solution. One measure was the DNase I hypersensitivity observed in footprint titrations in bands located between the two binding sites. The second measure resulted from the electrophoretic resolution in the gel mobility-shift titrations of the band representing the doubly liganded “tandem complex” from the band representing the singly liganded complexes, including the looped complex. Analysis of the footprint and mobility-shift titration data utilizing this additional information showed that approximately 65% of the molecules present in solution are looped complexes at pH 7.0, 100 mM KCl, and 20 °C when the binding sites on the DNA are saturated with protein. Reconciliation of the observed low binding cooperativity and the high proportion of looped complexes could only be obtained when the titration data were analyzed by a model in which Lac repressor tetramers dissociate into dimers in solution. The proportion of looped complexes present in solution is highly dependent on the dimer-tetramer association constant, ΔGtet. This result is consistent with the determination by high-pressure fluorescence techniques that Lac repressor tetramers dissociate with an association free energy comparable to their DNA-binding free energies [Royer, C. A., Chakerian, A. E., & Matthews, K. S. (1990) Biochemistry 29, 4959-4966]. However, when the value of ΔGtet of-10.6 kcal/mol (at 20 °C) reported by Royer et al. (1990) is assumed, the titration data demand that tetramers bind DNA with much greater affinity than dimers: a result inconsistent with the destabilization of tetramers by the operator observed in the dimer-tetramer dissociation studies. Analysis of the titration data subject to the assumption that the DNA-binding affinity of dimers and tetramers are identical yields a value of free energy of dimer-tetramer association, ΔGtet, of-13.2 ± 0.5 kcal/mol and a value of the “cyclization free energy”, ΔGj, of +12.2 ± 0.2 kcal/mol. Since the DNA-binding free energies of the Lac repressor to the 0EL and OIL binding sites are -13.8 and -13.7 ± 0.2 kcal/mol, respectively, the result is a looped complex that is only 10-15-fold more stable than the DNA-tetramer complex containing only a single protein-DNA interaction. The cyclization probability, or “j factor” [Shore, D., & Baldwin, R. L. (1983) J. Mol. Biol. 170, 957-981], calculated from ΔGj is approximately 0.8 × 10-9 for the intervening DNA in the looped complex. This probability corresponds to that predicted for the cyclization of a DNA fragment of approximately 16 helical turns (167 bp). This value is comparable, to a first approximation, with that expected for the cyclization probability of free DNA when the length of the bridging protein is included in the calculation of the size of the loop.
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