Elasticity Theory of a Twisted Stack of Plates

TL;DR

This paper develops an elastic model of DNA as a stack of rigid plates, deriving a twist-stretch energy with unique properties, revealing how DNA's structure influences its elastic response to stretching and twisting.

Contribution

It introduces a novel elastic model of DNA using a stack of rigid plates, deriving a twist-stretch energy with properties dependent on the plates' orientation and coupling constants.

Findings

The twist-stretch modulus vanishes when the helical radius is zero.

The modulus vanishes if certain elastic constants are zero and plates are perpendicular to the axis.

A laminated helical structure in an isotropic medium does not twist under stretch, but a bent one does.

Abstract

We present an elastic model of B-form DNA as a stack of thin, rigid plates or base pairs that are not permitted to deform. The symmetry of DNA and the constraint of plate rigidity limit the number of bulk elastic constants contributing to a macroscopic elasticity theory of DNA to four. We derive an effective twist-stretch energy in terms of the macroscopic stretch epsilon along and relative excess twist sigma about the DNA molecular axis. In addition to the bulk stretch and twist moduli found previously, we obtain a twist-stretch modulus with the following remarkable properties: 1) it vanishes when the radius of the helical curve following the geometric center of each plate is zero, 2) it vanishes with the elastic constant K_{23} that couples compression normal to the plates to a shear strain, if the plates are perpendicular to the molecular axis, and 3) it is nonzero if the plates are tilted relative to the molecular axis. This implies that a laminated helical structure carved out of an isotropic elastic medium will not twist in response to a stretching force, but an isotropic material will twist if it is bent into the shape of a helix.