Rawdon Eric J, Millett Kenneth C, Stasiak Andrzej
University of St. Thomas, Department of Mathematics, Saint Paul, MN, USA.
University of California Santa Barbara, Department of Mathematics, Santa Barbara, CA, USA.
Sci Rep. 2015 Mar 10;5:8928. doi: 10.1038/srep08928.
We introduce disk matrices which encode the knotting of all subchains in circular knot configurations. The disk matrices allow us to dissect circular knots into their subknots, i.e. knot types formed by subchains of the global knot. The identification of subknots is based on the study of linear chains in which a knot type is associated to the chain by means of a spatially robust closure protocol. We characterize the sets of observed subknot types in global knots taking energy-minimized shapes such as KnotPlot configurations and ideal geometric configurations. We compare the sets of observed subknots to knot types obtained by changing crossings in the classical prime knot diagrams. Building upon this analysis, we study the sets of subknots in random configurations of corresponding knot types. In many of the knot types we analyzed, the sets of subknots from the ideal geometric configurations are found in each of the hundreds of random configurations of the same global knot type. We also compare the sets of subknots observed in open protein knots with the subknots observed in the ideal configurations of the corresponding knot type. This comparison enables us to explain the specific dispositions of subknots in the analyzed protein knots.
我们引入了圆盘矩阵，它对圆形纽结构型中所有子链的纽结情况进行编码。圆盘矩阵使我们能够将圆形纽结分解为其子纽结，即由全局纽结的子链形成的纽结类型。子纽结的识别基于对线性链的研究，其中通过空间稳健的封闭协议将纽结类型与链相关联。我们通过采用能量最小化形状（如KnotPlot构型和理想几何构型）来表征全局纽结中观察到的子纽结类型集。我们将观察到的子纽结集与通过改变经典素纽结图中的交叉点得到的纽结类型进行比较。基于此分析，我们研究了相应纽结类型的随机构型中的子纽结集。在我们分析的许多纽结类型中，在相同全局纽结类型的数百个随机构型中的每一个中都发现了来自理想几何构型的子纽结集。我们还将在开放蛋白质纽结中观察到的子纽结集与在相应纽结类型的理想构型中观察到的子纽结进行比较。这种比较使我们能够解释所分析的蛋白质纽结中子纽结的具体排列方式。
