Macromolecules In The Elastic Rod Model, The Bead Spring Model And Its Numerical Simulation

The dynamics of DNA macromolecule is an important subject in biomechanics which has drawn great attention in recent years. Elasic rod and bead-spring are two important models for analyzing its dynamical properties. Since long-chain molecules have Macro-structural characteristics in length and nanoscale in diameters have microscopic nature, many knowledges in different fields such as elasticity, Brownian dynamics and molecule biology are combined to study its modeling, numerical simulation and dynamical analysis, and significant progresses have been made recently.In this paper, the configuration space of elastic rod is studied. The Lax pair for SMK equation with discrete time is obtained for Kirchhoff rod. FENE dumbbell model and Hookean dumbbell model and their numerical simulation are also involved. The main results in this paper are as follows:(1) At the macro scale, based on the Kirchhoff elastic rod model, the Lax pair for SMK equation with time discrete is discussed, and the semi-discrete Kirchhoff rod equation is obtained.(2) At the micro scale, based on dumbbell dynamic equations, Smoluchowsk equations for both Hookean model and FENE dumbbell model with internal viscosity have been solved. By using Ito formula, the equations are transformed into Ito stochastic differential equations, and then simulated with different parameters using Euler method. Finally, the numerical results for both models are analysised and compared.