Overcrossings

More generally, the metathesis reaction provides a laboratory analog of an important theorem in topology. Any linked or knotted structure can be converted to simple cycles (unknots) by selective interconversion of overcrossings and under-... 

Figure M-1. Example of calculation of the mean overcrossing number.

Arteca, G.A. (1999). Path-Integral Calculation of the Mean Number of Overcrossings in an Entangled Polymer Network. J.Chem.Inf.Comput.ScL, 39,550-557. 

Arteca, G.A. (1999) Path-integral calculation of the mean number of overcrossings in an entangled polymer network. /. Chem. Inf. Comput. Sci., 39, 550-557. 

The overcrossing probabilities for a molecule with at least three atoms... 

These descriptors capture essential folding features. Qualitatively, the basic properties are as follows for convoluted and entangled backbones, N and N mke large values and A is small and for swollen and disentangled chains, N — 0 and A — 1. Analyzed in detail, the overcrossing distribution allows... 

Figure 4 Comparison of backbones and overcrossing spectra for ribonuclease inhibitor (IBNH) and yeast hexokinase B (2YHX). These two proteins are similar in number of amino acid residues but radically different in their folding patterns. The difference is well reflected by the shape descriptor A, . The spectra are the superposition of five randomizations of projections. See Table 1.

Their different folding features become more evident upon comparison of the two overcrossing spectra (i.e., the histograms of An( ) ) in the lower part of Figure 4. [These spectra superimpose five computations with various numbers of randomized projections.] Without resorting to visual inspection, the shape descriptor indicates immediately that these two proteins have no 3D folding homology. 

The complete overcrossing spectra are contrasted in Figure 4 and 5. The statistical errors are estimated by computing five spectra for each backbone, with m = s x 10 randomized points, with s = 2, 3, 4, S, and 6. Each randomization was initialized with a different seed. See Ref. 94. 

This method of shape analysis applies also to other architectures. For instance, fluctuations in hydrogen-bonding networks play a key role in understanding solution properties. 3 Geometrical descriptors have been used to study these networks, including distribution of bonds and path connectivity in space.Overcrossing probabilities have been proposed as a tool for more detailed shape characterization,... 

One can design other shape descriptors that preserve the original geometric information while taking into account the handedness. For example, the handedness can be included in a colored overcrossing graph. 1 ... 

Similarly, we can extend the notion of overcrossing probabilities to include handedness. Consider a given 2D projection of the oriented polymer... 

Analysis of molecular loops is based on the features that differentiate between curves in space. In its simplest form, curves can be classified into two classes according to their possible embedding in the plane. A molecular chain that can, after allowed deformations, be drawn ( embedded ) in two dimensions is essentially different from one that cannot. In this latter case, all possible two-dimensional projections of the curve will exhibit overcrossings and never be reduced to a planar structure. 

Under these deformations, a closed circular loop will always remain closed and circular it will not be transformed into an object with different topological properties. Similarly, a topological knot cannot be untied by such a deformation. Therefore, these two curves will not be topologically equivalent. As shown next, some topological invariants of curves can be derived from their overcrossing properties. 

The most prominent shape features in these structures are described by topological invariants derived from knot theory. I3.i25.i26 Although the minimum number of overcrossings can be used as an invariant, i the more powerful and discriminating descriptors are the knot polynomials. These knot invariants are polynomials of a scalar t with rational coefficients. These functions translate in algebraic terms the basic topological features of space curves (or, more accurately, the space about them). They are related to the curve overcrossings, i ... 

There are two chiralities of trefoil knots right-handed and left-handed, depending on overcrossing handedness along the oriented curve (see Figure 6). These two knots are not distinguished by the polynomial A(t). For complicated knots, the computation of the entire polynomial is feasible, but it can be a formidable task. i... 

A more serious drawback of these polynomials is their poor discrimination for objects with a large minimum number of overcrossings. However, their performance can be improved by a number of alternatives. 3,127 "Jije Jones... 

Figure 6 Examples of simplest knots in molecular loops. The trefoil knot exhibits chirality. The handedness is determined by the right-hand rule. All three overcrossings in the (T+) knot have handedness of +1, whereas this is —1 in the case of the (T ) knot.

Figure 7 Monitoring structural stability in terms of shap fluctuations Comparison of the fluctuations in the mean number of overcrossings N(f) along molecular dynamics trajectories of deca(L-alanine) and deca(L-glycine). The computations correspond to isolated peptides immersed in a thermal bath at 500 K. The initial conformations were a-helices. See text for further details.

---