Knots in Globule and Coil Phases of a Model Polyethylene

Virnau, Peter; Kantor, Yacov; Kardar, Mehran

doi:10.1021/ja052438a

Cited by 169 publications

(245 citation statements)

References 48 publications

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“…Indeed, the analysis of protein conformations during early stages of folding clearly shows a noticeable number of randomly knotted structures. Such behavior agrees with results of simulations in (24) and well-known experimental results that flexible polymers or strings (25) can easily become spontaneously knotted. However, in most cases random knots observed in folding simulations do not lead to deep, but to relaxed knots.…”

Section: Discussionsupporting

confidence: 90%

Dodging the crisis of folding proteins with knots

Sułkowska

Sułkowski

Onuchic

2009

Proc. Natl. Acad. Sci. U.S.A.

167

217

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Proteins with nontrivial topology, containing knots and slipknots, have the ability to fold to their native states without any additional external forces invoked. A mechanism is suggested for folding of these proteins, such as YibK and YbeA, that involves an intermediate configuration with a slipknot. It elucidates the role of topological barriers and backtracking during the folding event. It also illustrates that native contacts are sufficient to guarantee folding in Ϸ1-2% of the simulations, and how slipknot intermediates are needed to reduce the topological bottlenecks. As expected, simulations of proteins with similar structure but with knot removed fold much more efficiently, clearly demonstrating the origin of these topological barriers. Although these studies are based on a simple coarse-grained model, they are already able to extract some of the underlying principles governing folding in such complex topologies. molecular dynamics ͉ slipknots ͉ backtracking ͉ topological barriers D uring the past 2 decades, a joint theoretical and experimental effort has largely advanced the quantitative understanding of the protein folding mechanism. Most small-and intermediate-size proteins live on a minimally frustrated funnel-like energy landscape, which allows fast and robust folding (1-3). Because proteins have been able to solve the energy problem, the final challenge is the structural complexity of the protein folding motifs. Most proteins avoid complex topologies, but recent discoveries have shown that some proteins are actually able to fold into nontrivial topologies where the main chain folds into a knotted conformation (4-6). Although these ''knotted'' folding motifs have been observed, we still have to face the challenging question of how the protein overcomes the kinetic barrier associated with the search of the knotted conformation. We suggest a possible mechanism where the knot formation is preceded by a conformation called a ''slipknot.'' A slipknot is topologically similar to a knot, except that an internal knot is effectively undone as the pathway of the backbone folds back on itself. The fact that such slipknots have already been observed in some protein final structures (7) adds support to this suggestion.This folding mechanism is explored in the context of the two most experimentally investigated knotted families of proteins, Haemophilus influenzae YibK and Escherichia coli YbeA, which are homodimeric ␣/␤-knot methyltransferases (MTases). A schematic representation of these proteins is shown in Fig. 1, [see also supporting information (SI) Fig. S1]. It has been shown experimentally that both these proteins unfold spontaneously and reversibly on addition of chemical denaturant (8-11) and they are able to fold even when additional domains are attached to one or both termini (12). In very recent experimental work (13), based on analysis of the effect of mutations in the knotted region of the protein, a folding model for YibK was also proposed. In this model the threading of the polypeptide chain and fo...

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Section: Discussionsupporting

confidence: 90%

Dodging the crisis of folding proteins with knots

Sułkowska

Sułkowski

Onuchic

2009

Proc. Natl. Acad. Sci. U.S.A.

167

217

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“…In the case of strings and homopolymers (22,23), there are no preferred locations for nucleation of knots, and knots are equally likely to be , and N. The minimum at low Q is the unfolded ensemble, the broad minimum at Q ≈ 0.2 corresponds to the formation of the β-sheet and correct twisting, and the broad minimum at high Q is the native ensemble. The red dotted line shows the probability P K of finding a knot as a function of Q.…”

Section: Resultsmentioning

confidence: 99%

Slipknotting upon native-like loop formation in a trefoil knot protein

Noel

Sułkowska

Onuchic

2010

Proc. Natl. Acad. Sci. U.S.A.

110

163

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Protein knots and slipknots, mostly regarded as intriguing oddities, are gradually being recognized as significant structural motifs. Recent experimental results show that knotting, starting from a fully extended polypeptide, has not yet been observed. Understanding the nucleation process of folding knots is thus a natural challenge for both experimental and theoretical investigation. In this study, we employ energy landscape theory and molecular dynamics to elucidate the entire folding mechanism. The full free energy landscape of a knotted protein is mapped using an all-atom structure-based protein model. Results show that, due to the topological constraint, the protein folds through a three-state mechanism that contains (i) a precise nucleation site that creates a correctly twisted native loop (first barrier) and (ii) a rate-limiting free energy barrier that is traversed by two parallel knot-forming routes. The main route corresponds to a slipknot conformation, a collapsed configuration where the C-terminal helix adopts a hairpin-like configuration while threading, and the minor route to an entropically limited plug motion, where the extended terminus is threaded as through a needle. Knot formation is a late transition state process and results show that random (nonspecific) knots are a very rare and unstable set of configurations both at and below folding temperature. Our study shows that a native-biased landscape is sufficient to fold complex topologies and presents a folding mechanism generalizable to all known knotted protein topologies: knotting via threading a native-like loop in a preordered intermediate.free energy landscape | knotted protein kinetics | nontrivial protein topology | protein folding | structure-based model P rotein structures have been observed with several complex folding motifs including knots and slipknots. These include nontrivial topologies containing 3 1 , 4 1 , 5 2 , and 6 1 knots (1-5). While the mechanism by which these proteins manage to reliably fold from a disordered linear polypeptide into complicated topologies is still largely a mystery, energy landscape theory is starting to provide us the key to resolve this challenge. In a minimally frustrated, funnel-like energy landscape, one expects that native contacts are on average favorable and dominate over nonfavorable nonnative ones (6-8). Topological constraints imposed by the existence of a native knot radically alters the funneled landscape. Many routes are barred from reaching the native state due to the obstacle imposed by the knot. Forming a knot requires intricate crossings of the polypeptide; any one made incorrectly leads to an unknotted protein or a wrong chirality. Therefore at first sight the problem of folding knots appears perplexing, but there is no reason to doubt that clues will be found in the native structure itself. Here it is shown how an all-atom structure-based model, which is dominated by native attractive interactions, is sufficient to uncover the energy landscape and folding routes of the smallest kn...

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“…14 For these reasons, there is a desire to elucidate the size and probabilities of knots in polymer molecules. Accordingly, simulations addressing this issue have been performed for numerous cases: linear 6,15 and circular 16 chains, ideal 16 and selfavoiding 6 chains, flexible 6,15,16 and semiflexible 17−20 chains, lattice 17,21 and off-lattice 17,19,20 models, good 6 and bad solvents, 15,22 as well as in free space, 17,19,20 in confinement, 23−25 and under tension. 19,20 An intriguing finding from simulations is that the cores of knots very often localize at small portion of chain.…”

Section: Introductionmentioning

confidence: 99%

“…19,20 An intriguing finding from simulations is that the cores of knots very often localize at small portion of chain. 15,16,21,23 For example, Katritch et al found the most probable size of trefoil knot on an unconfined circular ideal chain is only seven segments. 16 The localization of polymer knots has also observed in experiments, 26 and several theories have been developed to explain this behavior.…”

Section: Introductionmentioning

confidence: 99%

Metastable Tight Knots in Semiflexible Chains

2014

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Knotted structures can spontaneously occur in polymers such as DNA and proteins, and the formation of knots affects biological functions, mechanical strength and rheological properties. In this work, we calculate the equilibrium size distribution of trefoil knots in linear DNA using off-lattice simulations. We observe metastable knots on DNA, as predicted by Grosberg and Rabin. Furthermore, we extend their theory to incorporate the finite width of chains and show an agreement between our simulations and the modified theory for real chains. Our results suggest localized knots spontaneously occur in long DNA and the contour length in the knot ranges from 600 to 1800 nm.

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Knots in Globule and Coil Phases of a Model Polyethylene

Cited by 169 publications

References 48 publications

Dodging the crisis of folding proteins with knots

Dodging the crisis of folding proteins with knots

Slipknotting upon native-like loop formation in a trefoil knot protein

Metastable Tight Knots in Semiflexible Chains

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