A Generalized Theory of DNA Cyclization and Loop Formation

Abstract

We have developed a semi-analytic method for calculating the Stockmayer Jacobson J-factor for protein mediated DNA loops, as well as DNA ring cyclization. The formation of DNA loops on the order of a few persistence lengths is a key component in many gene regulatory functions. The binding of LacI protein within the Lac Operon of E.coli serves as the canonical example in which loop regulated transcription is understood. This fundamental looping motif consists of one protein simultaneously bound to two DNA operator binding sites. We explore as inputs the effect of sequence-dependent curvature and elasticity on the formation of DNA loops by constructing a Hamiltonian describing thermal fluctuations about the open and looped states. These fluctuations allow us to compute the entropic cost of loop formation, and thus allow a full computation of the free energy. Our work demonstrates that even for short sequences of the order one persistence length, entropic contributions are required to correctly compute the J factor.We determine the lowest energy shape of the inter-operator DNA loop using a non-linear mechanical rod model under prescribed binding topologies (e.g, parallel and anti-parallel binding). Expanding about this shape allows us to calculate the J factors associated with parallel and anti-parallel binding topologies within the Lac system, and thus how entropy influences the most energetically favorable topology. The J factor can be used to compare the relative loop lifetimes of various DNA sequences, making it a useful tool in sequence design. Our work also allows the computation of an effective torsional persistence length, which demonstrates how torsion bending coupling affects the conversion of writhe to twist.