Proteins are linear molecular chains that often fold to function. The topology of folding is widely believed to define its properties and function, and knot theory has been applied to study protein structure and its implications. More that 97% of proteins are, however, classified as unknots when intra-chain interactions are ignored. This raises the question as to whether knot theory can be extended to include intra-chain interactions and thus be able to categorize topology of the proteins that are otherwise classified as unknotted. Here, we develop knot theory for folded linear molecular chains and apply it to proteins. For this purpose, proteins will be thought of as an embedding of a linear segment into three dimensions, with additional structure coming from self-bonding. We then project to a two-dimensional diagram and consider the basic rules of equivalence between two diagrams. We further consider the representation of projections of proteins using Gauss codes, or strings of numbers and letters, and how we can equate these codes with changes allowed in the diagrams. Finally, we explore the possibility of applying the algebraic structure of quandles to distinguish the topologies of proteins. Because of the presence of bonds, we extend the theory to define bondles, a type of quandle particularly adapted to distinguishing the topological types of proteins.

中文翻译：

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蛋白质的结理论：高斯代码、quandles和bondles

蛋白质是线性分子链，通常会折叠以发挥功能。人们普遍认为折叠的拓扑结构定义了它的特性和功能，并且结理论已被应用于研究蛋白质结构及其含义。然而，当忽略链内相互作用时，超过 97% 的蛋白质被归类为未连接的蛋白质。这就提出了一个问题，即结理论是否可以扩展到包括链内相互作用，从而能够对被归类为未打结的蛋白质的拓扑结构进行分类。在这里，我们开发了折叠线性分子链的结理论并将其应用于蛋白质。为此，蛋白质将被认为是一个线性片段嵌入三个维度，附加结构来自自键。然后我们投影到二维图并考虑两个图之间等价的基本规则。我们进一步考虑使用高斯代码或数字和字母串表示蛋白质的投影，以及我们如何将这些代码与图表中允许的变化等同起来。最后，我们探索了应用 quandles 的代数结构来区分蛋白质拓扑结构的可能性。由于键的存在，我们扩展了理论来定义键，这是一种特别适用于区分蛋白质拓扑类型的乱码。我们探索了应用 quandles 的代数结构来区分蛋白质拓扑结构的可能性。由于键的存在，我们扩展了理论来定义键，这是一种特别适用于区分蛋白质拓扑类型的乱码。我们探索了应用 quandles 的代数结构来区分蛋白质拓扑结构的可能性。由于键的存在，我们扩展了理论来定义键，这是一种特别适用于区分蛋白质拓扑类型的乱码。