Numerical Solution for the Counterions Distribution in a Hexagonal DNA Lattice within Mean Field Theory Using Finite Element Method

Abstract

DNA has proven to be a promising material in fabrication and construction of complex structures with precise controlled nanoscale features. Most of the systems involving DNA are functioned in an aqueous solution where DNA molecules are strongly negatively charged. Therefore, understanding electrostatics of DNA system is essential for better understanding and design of DNA as a biomaterial and as a biological structure. In this work, the mean-field Poisson-Boltzmann equation for the distribution of mobile ions in a two-dimensional hexagonal lattice of DNA cylinders is solved numerically using finite element method. The weak formulation of the Poisson-Boltzmann equation is derived. The equation is then solved numerically using FreeFem++ scripting language. Our results show that the excess counterions of DNA dominate over the bulk ion concentration for the physiological salt concentration considered, and they condense on the DNA surface leading to very high charge density. The results also demonstrate the strong influence of the entropic confinement of the ions when the distance between neighboring DNA is smaller than 10 nm. This effect cannot be ignored in this case, and should be taken into account in any electrostatic investigation of DNA system.