﻿ DNA中一类非线性动力学方程的数值解 Numerical Solutions of a Kind of Nonlinear Dynamic Equations of the DNA

Vol.04 No.03(2015), Article ID:15937,8 pages
10.12677/AAM.2015.43034

Numerical Solutions of a Kind of Nonlinear Dynamic Equations of the DNA

Ting Li1, Bo Wang1,2

1School of Mathematics and Statistics, Henan University, Kaifeng Henan

2Institute of Applied Mathematics, Henan University, Kaifeng Henan

Email: wangbo_sdu@163.com

Received: Aug. 5th, 2015; accepted: Aug. 20th, 2015; published: Aug. 24th, 2015

ABSTRACT

We study nonlinear dynamics of DNA double helical under plane based on the rotator model proposed by Yomosa. Through dimensionless disposal, we get the more suitable data for numerical experiments and the relation between the data with the original parameters is also given. We solve a kind of nonlinear Sine-Gordon equations of the model by using the finite difference method, and the results show that the proposed numerical method is effective.

Keywords:DNA, Soliton, The Nonlinear Sine-Gordon Equations, The Dimensionless Disposal, The Finite Difference Method

DNA中一类非线性动力学方程的数值解

1河南大学数学与统计学院，河南 开封

2河南大学应用数学研究所，河南 开封

Email: wangbo_sdu@163.com

1. 引言

DNA即脱氧核糖核酸是生物遗传信息的负载者和遗传物质。在生命演化和生物体的生长和发育过程中承担着重要角色。近几年来的研究发现DNA动力学问题可以归结为研究非线性Sine-Gordon方程并找出其孤子解的问题。对于这类非线性方程，仅能得到某些特殊条件下的解析解。本文对一类非线性动力学方程无量纲化后进行数值计算，通过数值实验验证了所提出的差分格式对求解此类动力学方程的可行性。

2. 理论模型

1983年日本学者Yomosa提出的平面基转子模型[1] [2] 基本反映了DNA的结构和运动特征，此模型下系统Hamiltonian量可以表示为：

(1)

(2)

(3)

(4)

(5)

3. 非线性Sine-Gordon方程

(6)

(7)

4. 无量纲化

(8)

Table 1. Parameters in the model

Table 2. Comparison of model parameters

5. 数值实验

(9)

5.1. 差分格式

5.2. 局部截断误差

5.3. 数值计算结果

(a) (b)(c) (d)

Figure 1. The numerical solution of the scheme 1 (a), The analytic solution of Sine- Gordon equation (b), The absolute value of error (c), Positive and negative Kink soliton solution contrast of the scheme1 (d)

(a) (b)(c) (d)

Figure 2. The numerical solution of the scheme 1 (a), The analytic solution of Sine- Gordon equation (b), The absolute value of error (c), Positive and negative Kink soliton solution contrast of the scheme 1 (d)

Table 3. The resulting data of numerical experiment of scheme 1

Table 4. The resulting data of numerical experiment of scheme 2

Numerical Solutions of a Kind of Nonlinear Dynamic Equations of the DNA[J]. 应用数学进展, 2015, 04(03): 277-284. http://dx.doi.org/10.12677/AAM.2015.43034

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