Hu Shuangwei, Lundgren Martin, Niemi Antti J
Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 Jun;83(6 Pt 1):061908. doi: 10.1103/PhysRevE.83.061908. Epub 2011 Jun 15.
We develop a transfer matrix formalism to visualize the framing of discrete piecewise linear curves in three-dimensional space. Our approach is based on the concept of an intrinsically discrete curve. This enables us to more effectively describe curves that in the limit where the length of line segments vanishes approach fractal structures in lieu of continuous curves. We verify that in the case of differentiable curves the continuum limit of our discrete equation reproduces the generalized Frenet equation. In particular, we draw attention to the conceptual similarity between inflection points where the curvature vanishes and topologically stable solitons. As an application we consider folded proteins, their Hausdorff dimension is known to be fractal. We explain how to employ the orientation of C(β) carbons of amino acids along a protein backbone to introduce a preferred framing along the backbone. By analyzing the experimentally resolved fold geometries in the Protein Data Bank we observe that this C(β) framing relates intimately to the discrete Frenet framing. We also explain how inflection points (a.k.a. soliton centers) can be located in the loops and clarify their distinctive rôle in determining the loop structure of folded proteins.
我们开发了一种转移矩阵形式体系，以可视化三维空间中离散分段线性曲线的框架。我们的方法基于内在离散曲线的概念。这使我们能够更有效地描述在极限情况下线段长度消失时趋近分形结构而非连续曲线的曲线。我们验证了在可微曲线的情况下，我们离散方程的连续极限再现了广义弗伦内公式。特别地，我们提请注意曲率消失的拐点与拓扑稳定孤子之间的概念相似性。作为一个应用，我们考虑折叠蛋白质，已知它们的豪斯多夫维数是分形的。我们解释了如何利用氨基酸的C(β)碳原子沿着蛋白质主链的方向来引入沿着主链的优选框架。通过分析蛋白质数据库中实验解析的折叠几何结构，我们观察到这种C(β)框架与离散弗伦内框架密切相关。我们还解释了如何在环中定位拐点（又名孤子中心），并阐明它们在确定折叠蛋白质的环结构中的独特作用。
