Shape Descriptors of Macromolecular Topology

The geometrical descriptors just analyzed can be applied to chain molecules in both open and closed conformations. If we restrict the study to entanglements within (and between) loops, we can apply another group of shape descriptors. 

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. 

This example illustrates a more general, richer behavior. The topology of a space curve characterizes certain properties that remain invariant under ho-meomorphic transformations. These are continuous and reversible distortions, where the curve does not break, self-intersect, or join with other pieces. 

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 linking number can also be used as a shape descriptor of a single loop of double-stranded ( duplex ) DNA. In this case, the DNA is approximated by a continuous flat ribbon,and one considers the linking between the two curves defined by the borders of the ribbon (i.e., the two strands). The parameter measures the number of times the two strands intertwine around each other. In the DNA ribbon model, the linking number is relatedto the writhing number °lf of the central axis of the ribbon as 

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