Long Chain Macromolecules Brown Movement Mathematical Modeling And Numerical Methods

In recent years, many authors studied the dynamical properties of biological macromolecules such as DNA by using classical theory of dynamics as well as stochastic simulation. For example, the Kirchhoff theory on elastic rod has been used to analyze the supercoiled structure of DNA and the WLC model has been adopted to simulate the stochastic drift of a DNA molecular in liquid. Being a super slender elastic structure of 2nμin diameter and several centimeters in length, a DNA molecular shows both macro and micro properties while in motion. Therefore, a new dynamical model, benefited both from classical mechanics and stochastic simulation, is introduced in this paper to describe the dynamical properties of a DNA molecular. To raise the precision of the stochastic simulation, a new numerical approach is also introduced, and some techniques are introduced to reduce the complexity of the algorithm.The main results in this paper include two parts:(1) A numerical approach is raised to solve the stochastic differential equations for the WLC model. To obtain an algorithm of both high precision and efficient, the Ito-Taylor expansion is introduced to form a numerical method of strongly convergent of order 1, and then Fourier expansion is used to the diffusion term to simplify the computation. The hardest work in the algorithm is the computation the derivatives of matrix B at each time step since B, the Cholesky decomposition matrix of diffusion matrix D, could not be expressed in analytic form. However, benefited from its special structure, a compact computation process for computing (?)B/(?)x is raised in this paper.(2) To simulate the drift of DNA molecular in liquid, a new dynamical model is obtained by applying the random forces to the governing equations of an elastic rod. Arc length in the model is then discretized to form the semi-discrete scheme, and an algorithm of strongly convergent of order 1 is obtained by using Ito-Taylor expansion to the model. The model and the method provide a new approach to study the dynamical properties of a DNA macromolecular in the liquid environment.