Dynamics of charged molecular strands

Euler-Poincaré equations are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. The new feature is that the equations of motion for the dynamics of such molecular strands are nonlocal when the screened Coulomb interactions, or Lennard-Jones potentials between pairs of charges are included. These nonlocal dynamical equations are derived in the convective representation of continuum motion by using modified EulerPoincaré and Hamilton-Pontryagin variational formulations that illuminate the various approaches within the framework of symmetry reduction of Hamilton’s principle for exact geometric rods. In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods in the spatial representation. The motion equations in the convective representation are shown to be affine Euler-Poincaré equations relative to a certain cocycle. This property relates the geometry of the molecular strands to that of complex fluids. An elegant change of variables allows a direct passage from the infinite dimensional point of view to the covariant formulation in terms of Lagrange-Poincaré equations. In another revealing perspective, the convective representation of the nonlocal equations of molecular strand motion is transformed into quaternionic form. 1 Mathematics Department, Imperial College London, SW7 2AZ, UK [email protected], [email protected] Section de Mathématiques and Bernoulli Center, École Polytechnique Fédérale de Lausanne. CH–1015 Lausanne. Switzerland. [email protected], [email protected] 3 Institute for Mathematical Sciences, Imperial College London, SW7 2PG, UK 4 Department of Mathematics Colorado State University, Fort Collins, CO 80523-1874 [email protected] 5 Department of Mechanical Engineering, University of New Mexico, Albuquerque, NM 871311141 1 ar X iv :0 90 1. 29 59 v1 [ nl in .A O ] 1 9 Ja n 20 09 Ellis et al. Dynamics of charged molecular strands 2