Energy required to pinch a DNA plectoneme

DNA supercoiling plays an important role on a biological point of view. One of its consequences at the supra-molecular level is the formation of DNA superhelices named plectonemes. Normally separated by a distance on the order of 10 nm, the two opposite double-strands of a DNA plectoneme must be brought closer if a protein or protein complex implicated in genetic regulation is to be bound simultaneously to both strands, as if the plectoneme was locally pinched. We propose an analytic calculation of the energetic barrier, of elastic nature, required to bring closer the two loci situated on the opposed double-strands. We examine how this energy barrier scales with the DNA supercoiling. For physically relevant values of elastic parameters and of supercoiling density, we show that the energy barrier is in the $k_{\rm B} T$ range under physiological conditions, thus demonstrating that the limiting step to loci encounter is more likely the preceding plectoneme slithering bringing the two loci side by side.