It is known since the early days of molecular biology that proteins locate their specific targets on DNA up to two orders-of-magnitude faster than the Smoluchowski three-dimensional diffusion rate. An accepted explanation of this fact is that proteins are nonspecifically adsorbed on DNA, and sliding along DNA provides for the faster one-dimensional search. Surprisingly, the role of DNA conformation was never considered in this context. In this article, we explicitly address the relative role of three-dimensional diffusion and one-dimensional sliding along coiled or globular DNA and the possibility of correlated readsorption of desorbed proteins. We have identified a wealth of new different scaling regimes. We also found the maximal possible acceleration of the reaction due to sliding. We found that the maximum on the rate-versus-ionic strength curve is asymmetric, and that sliding can lead not only to acceleration, but also in some regimes to dramatic deceleration of the reaction.
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