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# Knotting and virtual knotting probabilities in different open curve ensembles. The closing distance fraction (CDF) is the ratio of the distance between the open curve's endpoints with respect to the total curve length. Knotting probabilities are given (a) 6 × 10 6 open random walks of length 100; (b) all 159,518 proteins analysed in the previous Section, with various lengths and binned according to CDF; (c) 5.5 × 10 6 length-75 subchains of Hamiltonian walks on cubic lattices of side length 6, binned by CDF. In (c), the large fluctuations reflect correlations implicit in the lattice. In each figure, the inset shows a typical example of the curve ensemble, coloured red to blue by hue along its length to distinguish different regions of the curve. Error bars represent the standard error on the mean probability of the knot statistic.

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Article
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Long, flexible physical filaments are naturally tangled and knotted, from macroscopic string down to long-chain molecules. The existence of knotting in a filament naturally affects its configuration and properties, and may be very stable or disappear rapidly under manipulation and interaction. Knotting has been previously identified in protein back...

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... fractal behaviour [38]. The probability of knotting in closed random walks has been well investigated [39]. Random walks tend not to be a good model for proteins, but nevertheless are good models for other physical systems [27,39,40], and are a convenient comparison model for knotting of open chains in the absence of physical constraints. Fig. 5(a) shows the statistics of knotting upon sphere and virtual closure for a set of random walks with 100 steps generated via the method of [41], with inset showing a sample random walk. The advantage of this particu- lar ensemble is that the CDF can be directly controlled, but for all distances knotting is significantly more common than ...
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... contrasts strongly with the equivalent results for proteins, shown in Fig. 5(b), which combine all protein chains from the previous Section despite their backbones being of many different lengths (from tens to thousands of angstroms and up to ∼3300 carbon atoms in the backbone chain). The comparatively small number of protein chains mean the statistics are only useful for qualitative compari- son. Nevertheless, ...
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... random walks, protein backbones are charac- terised by relatively compact geometries (such as the in- set to Fig. 5(b)), and aspects of this this can be repro- duced by simple mathematical models of random chains. In Fig. 5(c), we give the results for one such model: a subchain of a Hamiltonian walk [7], that is, a path on a cubic lattice of fixed size, visiting every vertex once and every edge no more than once. Such curves form a con- fined, folded ...
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... random walks, protein backbones are charac- terised by relatively compact geometries (such as the in- set to Fig. 5(b)), and aspects of this this can be repro- duced by simple mathematical models of random chains. In Fig. 5(c), we give the results for one such model: a subchain of a Hamiltonian walk [7], that is, a path on a cubic lattice of fixed size, visiting every vertex once and every edge no more than once. Such curves form a con- fined, folded structure due to the strict boundaries of the finite lattice. The geometry and topology of proteins are best ...
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... structure due to the strict boundaries of the finite lattice. The geometry and topology of proteins are best approximated when the Hamiltonian segment is much shorter than this, such that the lattice confinement is not strong, and random lattice walks of this type can be ef- ficiently generated up to lattice side lengths of at least 10 [42]. Fig. 5(c) shows the knotting and virtual knotting sam- pled from 5.5 × 10 6 random Hamiltonian subchains with length 75 on a cubic lattice of side length 6, with these parameters chosen to approximate the knotting probabili- ties in Fig. 5(b). Here the virtual knotting is strong rela- tive to closure knotting, comparable to proteins but very ...
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... random lattice walks of this type can be ef- ficiently generated up to lattice side lengths of at least 10 [42]. Fig. 5(c) shows the knotting and virtual knotting sam- pled from 5.5 × 10 6 random Hamiltonian subchains with length 75 on a cubic lattice of side length 6, with these parameters chosen to approximate the knotting probabili- ties in Fig. 5(b). Here the virtual knotting is strong rela- tive to closure knotting, comparable to proteins but very unlike random walks, and the probability of virtual knot- ting exceeds that of classical knotting across the small range 0.04 CDF 0.055. This trend appears to be highly robust to different parameters; even if the lattice is saturated, ...

## Citations

... In Figure 1, the fist two crossings are classical, and the third crossing is virtual. Virtual knots are used in the topological analysis of proteins since they naturally appear as virtual closers of knotoids; see [5][6][7]. One can define invariants of knotoids analogously to invariants of virtual knots [8]. ...
Article
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F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman’s affine index polynomial and use smoothing in the classical crossing of a virtual knot diagram. In this paper, we introduce weight functions for ordered orientable virtual and flat virtual links. A flat virtual link is an equivalence class of virtual links with respect to a local symmetry changing a type of classical crossing in a diagram. By considering three types of smoothing in classical crossings of a virtual link diagram and suitable weight functions, there is provided a recurrent construction for new invariants. It is demonstrated by explicit examples that newly defined polynomial invariants are stronger than F-polynomials.
... There are recently proposed models that do not involve closing the backbone, for example, virtual knots [1] and knotoids (open-knotted curves) [31,8,9,11,10]. ...
Article
Full-text available
We equip a knot K with a set of colored bonds, that is, colored intervals properly embedded into . Such a construction can be viewed as a structure that topologically models a closed protein chain including any type of bridges connecting the backbone residues. We introduce an invariant of such colored bonded knots that respects the HOMFLYPT relation, namely, the HOMFLYPT skein module of colored bonded knots. We show that the rigid version of the module is freely generated by colored Θ‐curves and handcuff links, while the nonrigid version is freely generated by the trivially embedded Θ‐curve. The latter module, however, does not provide information about the knottedness of the bonds.
... Despite advances, knot theory has limitations. The fraction of knotted proteins is only 0.77% of all proteins [24]. Also, to adhere with formal mathematical definitions, knots must be closed rings, which are rare in protein structures. ...
Article
The topology of biological polymers such as proteins and nucleic acids is an important aspect of their 3D structure. Recently, two applications of topology to molecular chains have emerged as important theoretical developments that are beginning to find utility in heteropolymer characterization and design: namely, circuit topology (CT) and knot theory. Here, we review the application of these two theories to protein, RNA, and DNA/genome structure, focusing on connections to conventional 3D structural information and relevance to function and highlighting recent experimental findings. We conclude with a discussion of recent applications to molecular origami and engineering.
... An important example is the Koniaris-Muthukumar-Taylor (KMT) algorithm developed in 2000 by Taylor (5), which extends to a wide range of protein conformations the application of the Koniaris-Muthukumar method (6) developed for ring polymers (Fig. 1C-D). Alternative methods to determine the knotting state of proteins based on different loop closure procedures have since then been proposed (7)(8)(9)(10)(11). More recently, the concept of knotoids introduced by Turaev (12) set the basis for a new method to study the entanglement of open protein chains (13), further refined in (14), and now available as an open access software tool (15). ...
... Independently of these methodological advances, the overall picture of the topological properties of the known native conformations is by now well established (16,17). The systematic application of knot detection methodologies based on the KMT algorithm to all available protein entries in the PDB revealed that about 1% correspond to knotted proteins (10,16). Although the trefoil is by large extent the most common knot type, it is possible to find a few proteins with more complex knots, including the Stevedore knot 6 1 (with six crossings on a planar projection) (18). ...
Chapter
A small fraction of all protein structures characterized so far are entangled. The challenge of understanding the properties of these knotted pro- teins, and the why and the how of their natural folding process, has been taken up in the past decade with different approaches, such as structural character- ization, in vitro experiments, and simulations of protein models with varying levels of complexity. The simplest among these are the lattice Go models, which belong to the class of structure-based models, i.e., models that are bi- ased to the native structure by explicitly including structural data. In this review we highlight the contributions to the field made in the scope of lattice Go models, putting them into perspective in the context of the main exper- imental and theoretical results and of other, more realistic, computational approaches.
... For proteins, the folded form determines function. Intriguingly, in spite of the dynamical challenge of folding, a fraction of the folded conformations of proteins are now known to have knots [3][4][5]. Interestingly, the structures in conventional polymer physics are distinct from those of biomolecules and include a coil (loosely analogous to an unfolded protein structure), an unstructured compact globule, rods assembled approximately parallel to each other, and a toroid. From a theoretical and modeling point of view, it is desirable to capture the essential attributes of a chain molecule in the simplest manner, while yet retaining the rich generic behavior of the diverse systems. ...
... The energy landscape has many low lying local minima and is not conducive to reproducible folding. In contrast, 3 1 knots are promoted almost exclusively by entropic stiffness and the ground state is much better separated in energy from the higher-energy states in accord with the observation [3][4][5] that more than 90% of protein knots are 3 1 in character. ...
Preprint
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Chain molecules play important roles in industry and in living cells. Our focus here is on distinct ways of modeling the stiffness inherent in a chain molecule. We consider three types of stiffnesses -- one yielding an energy penalty for local bends (energetic stiffness) and the other two forbidding certain classes of chain conformations (entropic stiffness). Using detailed Wang-Landau microcanonical Monte Carlo simulations, we study the interplay between the nature of the stiffness and the ground state conformation of a self-attracting chain. We find a wide range of ground state conformations including a coil, a globule, a toroid, rods, helices, zig-zag strands resembling $\beta$-sheets, as well as knotted conformations allowing us to bridge conventional polymer phases and biomolecular phases. An analytical mapping is derived between the persistence lengths stemming from energetic and entropic stiffness. Our study shows unambiguously that different stiffness play different physical roles and have very distinct effects on the nature of the ground state of the conformation of a chain, even if they lead to identical persistence lengths.
... There are recently proposed models that do not involve closing the backbone, for example, virtual knots [2] and knottoids (open-knotted curves) [31,12,13,11,10]. ...
... Note that the right-hand sides are connected sums of links with d−1 bonds and elementary generators from π c d , which can be cut off from the rest of the links by Equation (2). Note also that (l 4 + 2l 2 + 1 − l 2 m 2 ) = (l 2 + 1 − lm)(l 2 + 1 + lm) is invertible in R. ...
... Again, the elementary generators from the right-hand sides can be cut off using Equation (2). Proof. ...
Preprint
Full-text available
We can equip a knot $K$ with a set of colored bonds, that is, colored intervals properly embedded into $\mathbb{R}^3 \setminus K$. Such a construction can be viewed as a structure that topologically models a closed protein chain including any type of bridges connecting the backbone. We show that the HOMFLYPT skein module of (rigid-vertex) colored bonded links, defined in the usual way, is freely generated with an infinite basis consisting of colored $\Theta$-curves and handcuff links. A refined version of the invariant, together with computed examples, is given.
... Apart from classification with chain closure, proteins were also analyzed as virtual knots [116] and some exemplary knotted proteins were analyzes as knotoids [117,118]. In general, both classifications are consistent with the knot classification of the protein backbones. ...
... An important example is the Koniaris-Muthukumar-Taylor (KMT) algorithm developed in 2000 by Taylor (5), which extends to a wide range of protein conformations the application of the Koniaris-Muthukumar method (6) developed for ring polymers (Fig. 1C-D). Alternative methods to determine the knotting state of proteins based on different loop closure procedures have since then been proposed (7)(8)(9)(10)(11). More recently, the concept of knotoids introduced by Turaev (12) set the basis for a new method to study the entanglement of open protein chains (13), further refined in (14), and now available as an open access software tool (15). ...
... Independently of these methodological advances, the overall picture of the topological properties of the known native conformations is by now well established (16,17). The systematic application of knot detection methodologies based on the KMT algorithm to all available protein entries in the PDB revealed that about 1% correspond to knotted proteins (10,16). Although the trefoil is by large extent the most common knot type, it is possible to find a few proteins with more complex knots, including the Stevedore knot 6 1 (with six crossings on a planar projection) (18). ...
Preprint
A small fraction of all protein structures characterized so far are entangled. The challenge of understanding the properties of these knotted proteins, and the why and the how of their natural folding process, has been taken up in the past decade with different approaches, such as structural characterization, in vitro experiments, and simulations of protein models with varying levels of complexity. The simplest among these are the lattice Gō models, which belong to the class of structure-based models, i.e., models that are biased to the native structure by explicitly including structural data. In this review we highlight the contributions to the field made in the scope of lattice Gō models, putting them into perspective in the context of the main experimental and theoretical results and of other, more realistic, computational approaches.
... An important example is the Koniaris-Muthukumar-Taylor (KMT) algorithm developed in 2000 by Taylor (5), which extends to a wide range of protein conformations the application of the Koniaris-Muthukumar method (6) developed for ring polymers (Fig. 1C-D). Alternative methods to determine the knotting state of proteins based on different loop closure procedures have since then been proposed (7)(8)(9)(10)(11). More recently, the concept of knotoids introduced by Turaev (12) set the basis for a new method to study the entanglement of open protein chains (13), further refined in (14), and now available as an open access software tool (15). ...
... Independently of these methodological advances, the overall picture of the topological properties of the known native conformations is by now well established (16,17). The systematic application of knot detection methodologies based on the KMT algorithm to all available protein entries in the PDB revealed that about 1% correspond to knotted proteins (10,16). Although the trefoil is by large extent the most common knot type, it is possible to find a few proteins with more complex knots, including the Stevedore knot 6 1 (with six crossings on a planar projection) (18). ...
Preprint
A small fraction of all protein structures characterized so far are entangled. The challenge of understanding the properties of these knotted proteins, and the why and the how of their natural folding process, has been taken up in the past decade with different approaches, such as structural characterization, in vitro experiments, and simulations of protein models with varying levels of complexity. The simplest among these are the lattice G\=o models, which belong to the class of structure-based models, i.e., models that are biased to the native structure by explicitly including structural data. In this review we highlight the contributions to the field made in the scope of lattice G\=o models, putting them into perspective in the context of the main experimental and theoretical results and of other, more realistic, computational approaches.
... Although the first knotted protein was reported in 1977 1 , it was only after the development of knot detection methods 2, 3 , and loop closure procedures 4 , that these intricate systems came into the spotlight. Two recent surveys of the Protein Data Bank (PDB) indicate that the fraction of knotted proteins stands between 0.71% 5 and 0.77% 6 . The most frequent knot type found in the PDB is the 3 1 (or trefoil) (92.1%), followed by the 4 1 (4.8%), 5 2 (or twisted-three) (2.8%) and 6 1 (0.3%) 6 . ...
... Two recent surveys of the Protein Data Bank (PDB) indicate that the fraction of knotted proteins stands between 0.71% 5 and 0.77% 6 . The most frequent knot type found in the PDB is the 3 1 (or trefoil) (92.1%), followed by the 4 1 (4.8%), 5 2 (or twisted-three) (2.8%) and 6 1 (0.3%) 6 . ...
Article
There is growing support for the idea that the in vivo folding process of knotted proteins is assisted by chaperonins, but the mechanism of chaperonin-assisted folding remains elusive. Here, we conduct extensive Monte Carlo simulations of lattice and off-lattice models to explore the effects of confinement and hydrophobic intermolecular interactions with the chaperonin cage in the folding and knotting processes. We find that moderate to high protein-cavity interactions (which are likely to be established in the beginning of the chaperonin working cycle) cause an energetic destabilization of the protein that overcomes the entropic stabilization driven by excluded volume, and leads to a decrease of the melting temperature relative to bulk conditions. Moreover, mild-to-moderate hydrophobic interactions with the cavity (which would be established later in the cycle) lead to a significant enhancement of knotting probability in relation to bulk conditions while simultaneously moderating the effect of steric confinement in the enhancement of thermal stability. Our results thus indicate that the chaperonin may be able to assist knotting without simultaneously thermally stabilizing potential misfolded states to a point that would hamper productive folding thus compromising its functional role.