A Generalized Theory of DNA Looping and Cyclization

We have developed a generalized semi-analytic approach for efficiently computing cyclization and looping $J$ factors of DNA under arbitrary binding constraints. Many biological systems involving DNA-protein interactions impose precise boundary conditions on DNA, which necessitates a treatment beyond the Shimada-Yamakawa model for ring cyclization. Our model allows for DNA to be treated as a heteropolymer with sequence-dependent intrinsic curvature and stiffness. In this framework, we independently compute enthlapic and entropic contributions to the $J$ factor and show that even at small length scales $(\sim \ell_{p})$ entropic effects are significant. We propose a simple analytic formula to describe our numerical results for a homogenous DNA in planar loops, which can be used to predict experimental cyclization and loop formation rates as a function of loop size and binding geometry. We also introduce an effective torsional persistence length that describes the coupling between twist and bending of DNA when looped.