Conformational statistics of bent semiflexible polymers

Abstract

This paper extends previous methods for obtaining the probability distribution function of end-to-end distance for semiflexible polymers, and presents a general formalism that can generate conformational statistics of any continuum filament model of semiflexible chains with internal bends and twists. In particular, our focus is distribution functions for chains composed of straight or helical segments connected with discrete bends or twists. Prior polymer theories are not able to fully account for the effects of these internal shape discontinuities. We use the operational properties of the noncommutative Fourier transform for the group of rigid-body motions in three-dimensional space. This general method applies to various stiffness models of semiflexible chainlike macromolecules. Examples are given which apply the stiffness parameters defined in the Kratky-Porod model, Yamakawa helical wormlike chain model, and revised Marko-Siggia double-helix model to chains with intrinsic bends or twists in their undeformed ͑minimal energy͒ state. We demonstrate how the location and magnitude of internal bends in the chain affect the distribution of end-to-end distances for each of these models. This capability allows one to study the entropic effects of intrinsic shape changes ͑e.g., bend angle͒ in various models, and may lead to coarse-grained continuum mechanical models of processes that occur during transcription regulation.

Multiscale Modeling of Double-Helical DNA and RNA: A Unification through Lie Groups

“…Candidates for the form of b ( g ), which are a product of delta functions in position and shifted delta functions in orientation, were given by Zhou and Chirikjian. 75 This convolution approach is particularly useful when coupled with the fact that for the group Fourier transform defined in eq 16 $$false(\widehat{f_1\ast f_2}false)false(\lambda false)=\widehat{f_2}(\lambda )\widehat{f_1}(\lambda )$$…”

A unified approach to conformational statistics of classical polymer and polypeptide models

“…The PDF of end-to-end distance, denoted as f(r), is, in fact, a marginal 1-D PDF of the 6-D PDF of relative end-to-end position and orientation, denoted as F(g) where g = (r, R) ∈ SE (3). The final form of the result to be derived can be found in the literature [13,22,23]. However, since those do not contain detailed derivations, we derive it in this section.…”

“…With this in mind, we define the PDF of the i th link as (22) where g = (r, R(α, β, γ)) ∈ SE(3) and the partition function Z is defined as . The Fourier transform of this function becomes (23) Using the fact that vanishes except when j = −m, |m| ≤ l′, Eq. (23) is further simplified to the following form (24) only when |m| ≤ l′.…”

“…Recently, one unified methodology has been reported by which one can describe probability density functions with respect to all continuous models with quadratic energy function [22]. Also Zhou and Chirikjian have succeeded in generating probability density functions for bent semiflexible polymers with this general approach [23].…”

Conformational Statistics of Semiflexible Macromolecular Chains with Internal Joints

“…While classical semi-flexible polymer theories are in general inadequate to describe the conformational statistics of these systems, Rivetti et al made an effort to extend the applicability of the KP wormlike chain model to bent DNA, and derived closed-form expressions for the mean square end-to-end distance 58 . In our previous work, we proposed a general method to compute the full probability distribution of the end-to-end pose for the bent macromolecular chain with arbitrary stiffness and chirality parameters 59 . Ignoring the interactions between distal segments of the same macromolecular chain, our method applies to semi-flexible inextensible chiral elastic macromolecular chains with internal rigid bends and twists.…”