A new approach to the mechanics of DNA: Atoms-to-beam homogenization

Abstract It is useful to describe the deformation characteristics of long biological macromolecules, such as deoxyribonucleic acid (DNA), by means of terms such as “bending”, “stretching”, or “twisting”. These terms are borrowed from classical beam theory, a traditional and widely known subfield of continuum mechanics, whereas the standard numerical modeling procedure for macromolecules, which is molecular dynamics, does not allow for explicit introduction of the aforementioned deformation modes. This somehow puts some limit to the mechanical understanding of biological macromolecules. As a remedy, we here propose an upscaling (or homogenization) approach, spanning a new conceptual bridge from molecular dynamics to beam theory. Firstly, we apply the principle of virtual power (PVP) to classical continuum beams subjected to stretching and twisting, as well as to atomic compounds represented as discrete systems of mass points in the framework of molecular dynamics. Equating virtual power densities associated with continuum and discrete representations provides homogenization rules from the atomic compounds to the continuum beam line elements. Secondly, the forces acting on the aforementioned mass points are derived from energy potentials associated with bond stretching, valence and torsion angle variations, as well as electrostatic and van der Waals interactions. Application of this strategy to a specific DNA sequence consisting of 20 base pairs reveals deformation-dependent conformational changes, as well as paradox phenomena such as “stretching due to overwinding”, in line with known experimental observations.