Polymer topology

Random Knotting in Fractal Ring Polymers

Many ring polymer systems of physical and biological interest exhibit both pronounced topological effects and nontrivial self-similarity, but the relationship between these two phenomena has not yet been clearly established. Here, we use theory and simulation to formulate such a connection by studying a fundamental topological property—the random knotting probability—for ring polymers with varying fractal dimension, d_f. Using straightforward scaling arguments, we generalize a classic mathematical result, showing that the probability of a trivial knot decays exponentially with chain size, N, for all fractal dimensions: P₀(N) ∝ exp(−N/N₀). However, no such simple considerations can account for the dependence of the knotting length, N₀, on d_f, necessitating a more involved analytical calculation. This analysis reveals a complicated double-exponential dependence, which is well supported by numerical data. By contrast, functional forms typical of simple scaling theories fail to adequately describe the observations. These findings are equally valid for two-dimensional ring polymer systems, where “knotting” is defined as the intersection of any two segments.

Balancing Segregation and Complexation in Amphiphilic Copolymers by Architecture and Confinement

Segregation is a well-known principle for micellization, as solvophobic components try to minimize interactions with other entities (such as solvent) by self-assembly. An opposite principle is based on complexation (or coacervation), leading to the coassembly/association of different components. Most cases in the literature rely on only one of these modes, though the classical micellization scheme (such as spherical micelles, wormlike micelles, and vesicles) can be enriched by a subtle balance of segregation and complexation. Because of their counteraction, micellar constructs with unprecedented structure and behavior could be obtained. In this feature, systems are highlighted, which are between both mechanisms, and we study concentration, architecture, and confinement effects. Systems with inter- and intramolecular interactions are presented, and the effects of polymer topology and monomer sequence on the resulting structures are discussed. It is shown that complexation can lead to altered micellization behavior as the complex of one hydrophobic and one hydrophilic component can have a very low surface tension toward the solvent. Then, the more soluble component is enriched at the surface of the complex and acts as a microsurfactant. Although segregation dominates for amphiphilic copolymers in solution, the effect of the complexation can be enhanced by branching (change of architecture). Another possibility to enhance the complexation is by confining copolymers in a (pseudo-) 2D environment (like the one available at liquid-liquid interfaces). These observations show how new structural features can be achieved by tuning the subtle balance between segregation and complexation/solubilization.

Knotting pathways in proteins

Most proteins, in order to perform their biological function, have to fold to a compact native state. The increasing number of knotted and slipknotted proteins identified suggests that proteins are able to manoeuvre around topological barriers during folding. In the present article, we review the current progress in elucidating the knotting process in proteins. Although we concentrate on theoretical approaches, where a knotted topology can be unambiguously detected, comparison with experiments is also reviewed. Numerical simulations suggest that the folding process for small knotted proteins is composed of twisted loop formation and then threading by either slipknot geometries or flipping. As the size of the knotted proteins increases, particularly for more deeply threaded termini, the prevalence of traps in the free energy landscape also increases. Thus, in the case of longer knotted and slipknotted proteins, the folding mechanism is probably supported by chaperones. Overall, results imply that knotted proteins can be folded efficiently and survive evolutionary pressure in order to perform their biological functions.

Structures and folding pathways of topologically knotted proteins

In the last decade, a new class of proteins has emerged that contain a topological knot in their backbone. Although these structures are rare, they nevertheless challenge our understanding of protein folding. In this review, we provide a short overview of topologically knotted proteins with an emphasis on newly discovered structures. We discuss the current knowledge in the field, including recent developments in both experimental and computational studies that have shed light on how these intricate structures fold.

Identification of rare slipknots in proteins and their implications for stability and folding

Among the thousands of known three-dimensional protein folds, only a few have been found whose backbones are in knotted configurations. The rarity of knotted proteins has important implications for how natural proteins reach their natively folded states. Proteins with such unusual features offer unique opportunities for studying the relationships between structure, folding, and stability. Here we report the identification of a unique slipknot feature in the fold of a well-known thermostable protein, alkaline phosphatase. A slipknot is created when a knot is formed by part of a protein chain, after which the backbone doubles back so that the entire structure becomes unknotted in a mathematical sense. Slipknots are therefore not detected by computational tests that look for knots in complete protein structures. A computational survey looking specifically for slipknots in the Protein Data Bank reveals a few other instances in addition to alkaline phosphatase. Unexpected similarities are noted among some of the proteins identified. In addition, two transmembrane proteins are found to contain slipknots. Finally, mutagenesis experiments on alkaline phosphatase are used to probe the contribution the slipknot feature makes to thermal stability. The trends and conserved features observed in these proteins provide new insights into mechanisms of protein folding and stability.

Directed polymers with constrained winding angle

In this paper we study from a nonperturbative point of view the entanglement of two directed polymers subjected to repulsive interactions given by a Dirac delta-function potential. An exact formula of the so-called second moment of the winding angle is derived. This result is used to provide a thorough analysis of entanglement phenomena in the classical system of two polymers subjected to repulsive interactions and related problems. No approximation is made in treating the constraint on the winding angle and the repulsive forces. In particular, we investigate how repulsive forces influence the entanglement degree of the two-polymer system. In the limit of ideal polymers, in which the interactions are switched off, we show that our results are in agreement with those of previous works.