Shape of adsorbed supercoiled plasmids: an equilibrium description

Inspired by recent atomic force microscope (AFM) images of plasmids deposited on oppositely charged supported lipid bilayers from salt free solution, we propose a model for strongly adsorbed supercoiled cyclic stiff polyelectrolytes. We discuss how the excess linking number Lk of the deposited cycle is shared between writhe Wr and twist Tw at equilibrium and obtain the typical number of self-crossings in the deposited cycle as a function of surface charge density. The number of crossings at equilibrium is simply determined by the crossing penalty which is a local quantity and by the excess linking number. The number of crossings is well defined despite versatile plasmid shapes. For moderate numbers of crossings the loops are rather small and localized along the primary cycle, as expected from entropic loops. In the regime of many crossings, the cycle takes the shape of a regular flat ply ruled by local stiffness. The model allows for a semiquantitative comparison with the AFM images of deposited plasmids which are strongly charged.