Looping probabilities of elastic chains: a path integral approach.

Area of Science:

• Statistical Mechanics
• Polymer Physics
• Computational Biology

Background:

• DNA looping is critical for gene regulation.
• Existing models often simplify elastic chain properties.
• Accurate probability density functions (PDFs) are needed for DNA looping calculations.

Purpose of the Study:

• To derive an approximation to the PDF for elastic chain end-to-end locations and orientations.
• To develop a novel analytical method applicable to continuum mechanics models for DNA looping.
• To address limitations of prior approaches by not assuming chain uniformity or straight intrinsic shape.

Main Methods:

• Adoption of a path integral formalism.
• Calculation of a minimal energy configuration and a correction factor for quadratic fluctuations.
• Application of a nonlinear change of variable of Riccati type.
• Evaluation using a linear system of Jacobi ordinary differential equations.

Main Results:

• A novel approximation formula for the looping PDF is derived.
• The method successfully handles complex fluctuations with cross-terms in the Lagrangian.
• The Hamiltonian form of Jacobi equations provides corrections in the inextensible limit.
• An explicit, closed-form approximation is computed for a uniform circular arc chain.

Conclusions:

• The derived PDF approximation offers general applicability for elastic chains.
• The method provides a more accurate and versatile tool for DNA looping probability computations.
• This work advances analytical techniques in polymer physics and statistical mechanics.