Multi-scale multi-level homogenization for simulation of DNA molecules

The deformation of DNA has very important biological implication. The objective of this work is to develop a multi-scale coarse-graining method and a multi-level homogenization formulation for the effective numerical simulation of DNA at length-scales spanning from atomic to continuum. In the first part of this work, a coarse graining technique based on a multi-scale wavelet projection is proposed for modeling of DNA molecules. Based on the fine scale atomistic response and the Henderson's theorem, the distribution functions between centers of mass of two groups of atoms are employed to obtain the fine scale potential functions. These fine scale potential functions are then homogenized using the multi-scale wavelet projection to yield the coarse-scale effective potential functions between superatoms. Molecular dynamics simulation of DNA molecules under stretching process demonstrates that the results of fine scale model and the proposed coarse grained DNA model are in good agreement. The coarse grained simulation with a significant saving of computation time is shown to be in good agreement with the fine scale simulation. Due to the reduced spatial degrees of freedom and temporal discretization, a computational efficiency of three to four orders of magnitude is achieved in the proposed coarse grained model.;A second level homogenization of the coarse grained DNA is to construct an equivalent continuum strain energy density function of the DNA molecule. By further decomposing the total strain energy into the stretching, bending, and torsional energy of a beam, a continuum hyperelastic multi-scale beam formulation is obtained for continuum modeling of DNA molecules. Numerical tests on some of fundamental deformation modes of the atomistic model, coarse grained model, and continuum model are performed to validate the proposed methods. This continuum model is employed to study DNA loop formulation during protein-DNA interactions as well as DNA translocation through a nanopore under an electric force involved in applications such as DNA sequencing.