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... Introduc3on 1st par. ... tions with the DNA, but can o be rec gnized by the ta get site. I the activ conform tion, he pr t in is r cognized by the target site but has very rugged energy surface elsewher on the DNA. Following Zwanzig (29), diffusion n the rugged en rgy surface is equivalent to diffusion with a reduced, eff ctive diffusion constant on the smooth average energy surface. We denote the aver ge surf ce pote tials in the two conformations as UaðxÞ nd U iðxÞ. The effective diffusion constant for conformation g, g 1⁄4 a, or i, is Dg 1⁄4 D3e σg ; [16] where σg is the root-m an-squ re of h energy fluctuations aroun th smooth av age energy surface; the prefa tor could be significantly small r than D3 due to c pl d translational and rota ional diffusi n whe the protein moves along a h lical path while maintaining its orientation th respect to th DNA (30). Appar ntly, Slutsky and Mirny (12) have rediscovered Eq. 16. As illust ated in Fig. 1B, the a tive conformation has a much more rugg d en rgy surface t an the inactive conformation. C res ondingly, Da ≪ Di < D3. We assume that the switch between th tw conformati s is stochastic, with transition rates ωaðxÞ and ωiðxÞ (see Fig. 1B, SI Appendix: Fig. S1):