波形生物膜泡的数值法研究

Research of unduloid vesicles with numerical method

磷脂双亲分子在水溶液中会形成各种各样形状的膜泡,实验显示,存在有一维的周期性柱面膜泡。扩展的Delaunay曲面是由Ou-Yang等首次给出的Helfrich变分问题的解析特解,与Delaunay曲面不同的是它所代表的曲面为非常数平均曲率曲面,其中之一为波形周期曲面。本文用数值计算的方法探讨了波形曲面形状,并与已知的解析解进行了比较。与球形拓扑不同的是,做数值计算时所采用的欧拉-拉格朗日方程中参数不取为零,加入周期性边界条件数值求解该方程,得到了与扩展的Delaunay曲面一致的波形曲面。目前我们还没有得到扩展的Delaunay曲面之外的周期波形形状。扩展的Delaunay曲面是否给出了非常数平均曲率的波形曲面的通解,仍然是需要进一步探讨的问题。然后根据形状方程和轴对称的微分方程绘出了自发曲率取不同数值时的二维波形图,并且得出结论:随着自发曲率的增加,参数的值逐渐减小,与解析法得到的结果一致。

The phospholipid amphiphiles form various membrane vesicles in aqueous solution. The experiment shows that one-dimensional periodic cylinder is exist. The extended Delaunay equation is a particular solution of the Helfrich variational equation given by Ou-Yang etc. at the first time. It represents a class of surfaces with nonconstant mean curvature,one of which is the periodic unduloid surface. This paper discusses the periodic unduloid surface. The parameter different to the sphere topological shape in the Euler-Lagrange equations is not equal to zero. When adding the periodic boundery conditions to the Euler-Lagrange equations, we obtain the unduloid shape consisted with the extended Delaunay surface. We haven＇t got an unduloid shape beyond the extended Delaunay surface at present. This is a problem we discuss later. Then a two-dimensional unduloid figure with different spontaneous-curvature coefficient CO is ploted. With the increase of CO ,the parameter decreases. The results are consistent with those obtained from the analytical method.