Revealing Topological Barriers against Knot Untying in Thermal and Mechanical Protein Unfolding by Molecular Dynamics Simulations

The knot is one of the most remarkable topological features identified in an increasing number of proteins with important functions. However, little is known about how the knot is formed during protein folding, and untied or maintained in protein unfolding. By means of all-atom molecular dynamics simulation, here we employ methyltransferase YbeA as the knotted protein model to analyze changes of the knotted conformation coupled with protein unfolding under thermal and mechanical denaturing conditions. Our results show that the trefoil knot in YbeA is occasionally untied via knot loosening rather than sliding under enhanced thermal fluctuations. Through correlating protein unfolding with changes in the knot position and size, several aspects of barriers that jointly suppress knot untying are revealed. In particular, protein unfolding is always prior to knot untying and starts preferentially from separation of two α-helices (α1 and α5), which protect the hydrophobic core consisting of β-sheets (β1–β4) from exposure to water. These β-sheets form a loop through which α5 is threaded to form the knot. Hydrophobic and hydrogen bonding interactions inside the core stabilize the loop against loosening. In addition, residues at N-terminal of α5 define a rigid turning to impede α5 from sliding out of the loop. Site mutations are designed to specifically eliminate these barriers, and easier knot untying is achieved under the same denaturing conditions. These results provide new molecular level insights into the folding/unfolding of knotted proteins.

Phosphorescent Organometallic Knots of Gold(I)-Bis(acetylide) Strands Directed by Copper(I) π-Coordination

Through elaborate ligand design to create knotted structures with specific topologies is a major challenge for chemists. In this work, the self-assembly between U-shape 3,6-di-tert-butyl-1,8-diethynyl-9H-carbazole (H2L) and Au+ through gold(I)-bis(acetylide) linkages under π-bonded Cu+ template gives rise to complex 1 with two interlocked metallostrands as well as complexes 3 (n = 3) and 4 (n = 4) with [(AuL)n]n- metallostrands showing trefoil knot topology. Upon incorporating two [Au(dppb)Au]2+ (dppb = Ph2P(CH2)4PPh2) moieties through bis(Au-acetylide) coordination bonds, the interlocked structure (1) is fully closed to form a figure-eight knotted structure in complex 2. The folding and threading of metallocyclic strings are directed by Cu+, which are π-ligated to two or three acetylides to generate double-folding or triple-folding cross points. Complexes 1-4 show intense phosphorescence in both solutions and solid states at ambient temperature, originating from admixture of metal centered 3[d®p/s], 3IL (intraligand), and 3[p (L) ® s/p (Au/Cu)] 3LMCT triplet states.

A specific set of heterogeneous native interactions yields efficient knotting in protein folding

Native interactions are crucial for folding, and non-native interactions appear to be critical for efficiently knotting proteins. Therefore, it is important to understand both their roles in the folding of knotted proteins. It has been proposed that non-native interactions drive the correct order of contact formation, which is essential to avoid backtracking and efficiently self-tie. In this study we ask if non-native interactions are strictly necessary to tangle a protein, or if the correct order of contact formation can be assured by a specific set of native, but otherwise heterogeneous, interactions. In order to address this problem we conducted extensive Monte Carlo simulations of lattice models of proteinlike sequences designed to fold into a pre-selected knotted conformation embedding a trefoil knot. We were able to identify a specific set of heterogeneous native interactions that drives efficient knotting, and is able to fold the protein when combined with the remaining native interactions modeled as homogeneous. This specific set of heterogeneous native interactions is strictly enough to efficiently self-tie. A distinctive feature of these native interactions is that they do not backtrack, because their energies ensure the correct order of contact formation. Furthermore, they stabilize a knotted intermediate state, which is en-route to the native structure. Our results thus show that - at least in the context of the adopted model - non-native interactions are not necessary to knot a protein. However, when they are taken into account into protein energetics it is possible to find specific, non-local non-native interactions that operate as a scaffold that assists the knotting step.

On topology and knotty entanglement in protein folding

We investigate aspects of topology in protein folding. For this we numerically simulate the temperature driven folding and unfolding of the slipknotted archaeal virus protein AFV3-109. Due to knottiness the (un)folding is a topological process, it engages the entire backbone in a collective fashion. Accordingly we introduce a topological approach to model the process. Our simulations reveal that the (un)folding of AFV3-109 slipknot proceeds through a folding intermediate that has the topology of a trefoil knot. We observe that the final slipknot causes a slight swelling of the folded AFV3-109 structure. We disclose the relative stability of the strands and helices during both the folding and unfolding processes. We confirm results from previous studies that pointed out that it can be very demanding to simulate the formation of knotty self-entanglement, and we explain how the problems are circumvented: The slipknotted AFV3-109 protein is a very slow folder with a topologically demanding pathway, which needs to be properly accounted for in a simulation description. When we either increase the relative stiffness of bending, or when we decrease the speed of ambient cooling, the rate of slipknot formation rapidly increases.

Selective construction and stability studies of molecular trefoil knot and Solomon link

Two novel compounds, a molecular trefoil knot and a Solomon link, were constructed successfully through the cooperation of multiple π-π stacking interactions. A reversible transformation between the trefoil knot and...

Topological Assembly of a Deployable Hoberman Flight Ring from DNA

ABSTRACTDeployable geometries are finite auxetic structures that preserve their overall shapes during expansion and contraction. The topological behaviors emerge from intricately arranged elements and their connections. Despite considerable utility of such configurations in nature and in engineering, deployable nanostructures have never been demonstrated. Here we show a deployable flight ring, a simplified planar structure of Hoberman sphere, using DNA origami. The DNA flight ring consists of topologically assembled six triangles in two layers that can slide against each other, thereby switching between two distinct (open and closed) states. The origami topology is a trefoil knot, and its auxetic reconfiguration results in negative Poisson’s ratios. This work shows the feasibility of deployable nanoarchitectures, providing a versatile platform for topological studies and opening new opportunities for bioengineering.

Comparison of Fused Deposition Modeling and Color Jet 3D Printing Technologies for the Printing of Mathematical Geometries

Many mathematical geometries act as an optimal structure for functional applications and have always been an area of interest in the research field. Their topology offers properties which are crucial and can be used effectively in various domains. Apart from that, some have a resemblance to naturally occurring compounds which can help us to study their different transformations and behavior. In this paper, we present two such geometries, first, gyroid, which is an iso-minimal surface and second a three-crossing knot, also known as trefoil knot. The structure of gyroid makes it unique and is considered suitable in developing energy-absorbing, structural and lightweight applications. Similarly, some types of knots resemble the DNA structure and have found use in molecular chemistry. This paper discusses different application areas of these geometries. Further, this paper presents modeling and printing by using fused deposition modeling (FDM) and color jet printing (CJP). Comparative analysis has been done by considering various parameters. This paper discusses the potential of these two rapid prototyping technologies and their suitability for specific printing applications.

Suciu’s ribbon 2-knots with isomorphic group

Suciu constructed infinitely many ribbon 2-knots in [Formula: see text] whose knot groups are isomorphic to the trefoil knot group. They are distinguished by the second homotopy groups. We classify these knots by using [Formula: see text]-representations of the fundamental groups of the 2-fold branched covering spaces.

Untangle problems, with knot theory

“Untangle problems, with knot theory” offers a basic introduction to the mathematical subfield of knot theory, including the classification of knots by crossing numbers. A mathematical knot is a closed loop that may or may not be tangled. Two knots are considered the same if one may be manipulated into the other using easy-to-understand techniques. Readers learn to identify knots by crossing numbers and encounter numerous hand-drawn sketches of knots, including the trivial knot, trefoil knot, figure-eight knot, and more. Mathematics students and enthusiasts are encouraged to employ knot theory methods for untangling problems in mathematics or life by asking whether they have encountered the problem before. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.