Linear Random Knots and Their Scaling Behavior

Abstract

We present here a nonbiased probabilistic method that allows us to consistently analyze knottedness of linear random walks with up to several hundred noncorrelated steps. The method consists of analyzing the spectrum of knots formed by multiple closures of the same open walk through random points on a sphere enclosing the walk. Knottedness of individual "frozen" configurations of linear chains is therefore defined by a characteristic spectrum of realizable knots. We show that in the great majority of cases this method clearly defines the dominant knot type of a walk, i.e., the strongest component of the spectrum. In such cases, direct end-to-end closure creates a knot that usually coincides with the knot type that dominates the random closure spectrum. Interestingly, in a very small proportion of linear random walks, the knot type is not clearly defined. Such walks can be considered as residing in a border zone of the configuration space of two or more knot types. We also characterize the scaling behavior of linear random knots.

Microscopic Description of Entanglements in Polyethylene Networks and Melts: Strong, Weak, Pairwise, and Collective Attributes

“…The knots were defined at the real chain level by using as topological criterion a local form of the Gauss linking number. For open chains this criterion is not more rigorous 54,55,122 than the one defined here, while it is computationally more complex.…”

Conservation of complex knotting and slipknotting patterns in proteins

“…Of course, the characterization of the knotting within the protein structures makes sense only if one considers fixed configurations, in this case proteins in their native folded structures. Then they may be treated as frozen and thus unable to undergo any deformation.Several papers have described various interesting closure procedures to capture the knot type of the native structure of a protein or a subchain of a closed chain (1,3,(29)(30)(31)(32)(33). In general, the strategy is to ensure that the closure procedure does not affect the inherent entanglement in the analyzed protein chain or subchain.…”

“…However, the shortening method applied to the same starting configuration can result in different knot types depending on the order of the shortening moves (30). Because the order of the shortening moves is not determined by the actual configuration but depends on arbitrarily chosen parameters, this method also is confronted with the problem that the linear chains do not totally determine the knot type of the frozen chain.Limitations in these single closure methods stimulated interest in probabilistic methods of defining the most likely knot type of linear chains with a given geometry (1,3,30). One relatively simple, unbiased method consists of placing the analyzed linear chain near the center of a large sphere and closing it by adding to each end one long segment connecting it with the same, randomly chosen point on the enclosing sphere.…”

“…It is then easy to connect the ends without passing through the entangled portion. However, the shortening method applied to the same starting configuration can result in different knot types depending on the order of the shortening moves (30). Because the order of the shortening moves is not determined by the actual configuration but depends on arbitrarily chosen parameters, this method also is confronted with the problem that the linear chains do not totally determine the knot type of the frozen chain.…”

“…Limitations in these single closure methods stimulated interest in probabilistic methods of defining the most likely knot type of linear chains with a given geometry (1,3,30). One relatively simple, unbiased method consists of placing the analyzed linear chain near the center of a large sphere and closing it by adding to each end one long segment connecting it with the same, randomly chosen point on the enclosing sphere.…”