Self-knotting

A rescaling of D(Q) with respect to a length scale ls=(d Qy the mean distance between two successive self-knots where dj=2Q A is one adjustable parameter leads to the good superposition shown in Fig. 6.27. 

The relationship between the linking number and the helicity of a self-knotted filamental tube has been studied by Berger and Field [50] and Moffatt and Ricca [51]. We are looking for an expression for the helicity in a filamental tube 7 of strength 5, around a closed magnetic line C that can be self-knotted. In this case, the magnetic helicity takes the value... 

Using this theorem, we conclude that the magnetic helicity of a filamental tube T whose axe is a closed, self-knotted curve C, is given by... 

The dependence p( N) of the chain self-knotting probability p is determined as a function of chain length N by the random chain closure [29-31, 34]. In a recent work [36], the simulation procedure was extended up to the chains of order N 2000, where for trivial knot formation probability the exponential asymptotics of the type... 

The result of annealing on different kinds of false cyclic polymers (CPs) (a) valid, (b) self-knotted (trefoil knot—lattice representation of the same knot in Figure Zld). After armealing, false CPs that are a trivial knot attain a contour length of at most 3y[2-... 

Based on the state of knotting and catenation, the false CPs thus obtained were classified as either of (a) valid, V, (b) only self-knotted, S, (c) only catenated, C, or (d) both self-knotted and catenated, SC. These categories correspond to the regions shown schematically in Figure 7.13. The superscripts V, S, C, and SC are used to distinguish each class of false CPs, and the superscript A is used to designate the superset of all the false CPs (i.e., = P u S u... 

The average distance between beads of the four different classes of false cyclic polymers (CPs) for N = 150. As expected, the catenated, but otherwise unknotted, false CPs are the largest, and the self-knotted false CPs are the smallest. This plot is representative of the trend observed over all values of N andf. ... 

Arguments presented in Section 7.7 lead to a cubic dependence on molecular weight. More subtle effects involving local fluctuations and self-knots in chains must be invoked to augment the molecular-weight dependence. 

A useful summary of the various and numerous types of rotaxanes, catenanes, and knots can be found in a review of template routes to interlocked molecular structures 468). Inorganic chemistry is centrally involved in the templating involved in self-assembly and in controlled synthesis of such species. 

Another successful example of exploiting FACIT cystine knot was made on type XII collagen. A minichain consisting of COLl and the N-terminal portion of the NCI domain has been successfully self-assemhled into a triple helical homotrimer stabilized by the cystine knot." ... 

The simplest example of topological classification is the knotting of individual strands, i.e., self-entanglement. However, the effects attributed to entanglement in non-crosslinked polymers are clearly intermolecular. The simplest such case is that of pair-wise classification without self-entanglement. Consider the following three examples of strand pairs ... 

Inorganic templating and self-assembly provide coordination compounds whose geometries make possible the synthesis of complex structures, namely of cyclic multiporphyrin arrays [9.13a, 9.179], of inorganic rotaxanes [9.97a, 9.180], of multi-catenates and catenands (see 181) [8.281, 8.282] and even of molecular knots (see 182) [8.282, 9.77, 9.181] (in 181 and 182 a) with, b) without Cu(l) template). 

This process involves the covalent locking in of structures formed by reversible self-assembly. The irreversible, post-assembly step switches off the equilibrium process involved in the self-assembly. As we will see in the following sections, self assembly with covalent postmodification is involved in a range of biochemistry (e.g. insulin synthesis) and elegant abiotic supramolecular synthesis as in the formation of catenanes and knots. 

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