  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.
