Mean-square radius of gyration and the hydrodynamic radius for topological polymers expressed with graphs evaluated by the method of quaternions revisited

Programmed folding into spiro-multicyclic polymer topologies from linear and star-shaped chains

The development of precise folding techniques for synthetic polymer chains that replicate the unique structures and functions of biopolymers has long been a key challenge. In particular, spiro-type (i.e., 8-, trefoil-, and quatrefoil-shaped) polymer topologies remain challenging due to their inherent structural complexity. Herein, we establish a folding strategy to produce spiro-type multicyclic polymers via intramolecular ring-opening metathesis oligomerization of the norbornenyl groups attached at predetermined positions along a synthetic polymer precursor. This strategy provides easy access to the desired spiro-type topological polymers with a controllable number of ring units and molecular weight while retaining narrow dispersity (Ɖ < 1.1). This effective strategy marks an advancement in the development of functionalized materials composed of specific three-dimensional nanostructures.

Statistical Properties of Lasso-Shape Polymers and Their Implications for Complex Lasso Proteins Function

The shape and properties of closed loops depend on various topological factors. One of them is loop-threading, which is present in complex lasso proteins. In this work, we analyze the probability of loop-threading by the tail and its influence on the shape of the loop measured by the radius of gyration, distention, asphericity, and prolateness. In particular, we show that the probability of a trivial lasso for phantom polymer is non-zero even for an infinite structure, as well as that the threading flattens the loop by restricting its motion in one dimension. These results are further used to show that there are fewer non-trivial protein lassos than expected and select potentially functional complex lasso proteins.