Looping charged elastic rods: applications to protein-induced DNA loop formation

We analyze looping of thin charged elastic filaments under applied torques and end forces, using the solution of linear elasticity theory equations. In application to DNA, we account for its polyelectrolyte character and charge renormalization, calculating electrostatic energies stored in the loops. We argue that the standard theory of electrostatic persistence is only valid when the loop's radius of curvature and close-contact distance are much larger than the Debye screening length. We predict that larger twist rates are required to trigger looping of charged rods as compared with neutral ones. We then analyze loop shapes formed on charged filaments of finite length, mimicking DNA looping by proteins with two DNA-binding domains. We find optimal loop shapes at different salt amounts, minimizing the sum of DNA elastic, DNA electrostatic, and protein elastic energies. We implement a simple model where intercharge repulsions do not affect the loop shape directly but can choose the energy-optimized shape from the allowed loop types. At low salt concentrations more open loops are favored due to enhanced repulsion of DNA charges, consistent with the results of computer simulations on formation of DNA loops by lac repressor. Then, we model the precise geometry of DNA binding by the lac tetramer and explore loop shapes, varying the confined DNA length and protein opening angle. The characteristics of complexes obtained, such as the total loop energy, stretching forces required to maintain its shape, and the reduction of electrostatic energy with increment of salt, are in good agreement with the outcomes of more elaborate numerical calculations for lac-repressor-induced DNA looping.