Topological stereochemistry

The chirality of ground DNA knots and links

The chirality of ground DNA knots and links is described and characterized in terms of color symmetry groups (CSG), i.e. color symmetry groups I and II, which correspond to topochirality (topological chirality) and topoachirality (topological achirality) which bear an uncanny resemblance to point groups I (proper) and point groups II (improper) used for testing geochirality (geometrical chirality) and geoachirality (geometrical achirality), respectively. By regarding these two crossing modes in mirror images as white and black vertices, DNA knots and links with minimal crossings can be mapped to vertex-bicolored graphs under a working hypothesis that DNA knots and links exist in ground states with minimal energy m0. The color symmetry group of a vertex-bicolored graph G is defined as the set of all permutations and permutation asymmetrizations of the vertices of G that preserve its topology (connectivity), where asymmetrization, denoted as (a), is the operation of changing vertices' colors, and a permutation followed by an (a) is a permutation asymmetrization. The color symmetry groups I contains only permutations, whereas color symmetry groups II comprise permutation asymmetrizations as well as permutations. Four DNA knots and links in nature are analyzed and tabulated consisely. In addition, the well-known figure-of-eight knot and Borromean rings are discussed in much the same way.

Implementing knot-theoretical characterization methods to analyze the backbone structure of proteins: application to CTF L7/L12 and carboxypeptidase A inhibitor proteins

In this work we apply a recently developed method for characterizing the shape of the tertiary structure of proteins. The approach is based on a combination of graph- and knot-theoretical characterizations of Cartesian projections of the space curve describing the protein backbone. The proposed technique reduces the essential shape features to a topologically based code formed by a sequence of knot symbols and polynomials. These polynomials are topological invariants that describe the overcrossing and knotting patterns of curves derived from the molecular space curve. These descriptors are algorithmically computed. The procedure is applied to describe the structure of the carboxy terminal fragment of the L7/L12 chloroplast ribosomal protein (CTF L7/L12) and the potato carboxypeptidase A inhibitor protein (PCI), which has a set of three disulfide bridges. In the former case, we describe the protein's shape features in terms of its alpha-helices, and a backbone simplified by considering helices without internal structure. An extension of the methodology to describe disulfide bridges is discussed and applied to PCI. Changes in the knot-theoretical characterization due to possible uncertainties in the resolution of the X-ray structure, as well as the inclusion of low-frequency motions of the backbone, are also discussed.

A method for the characterization of foldings in protein ribbon models

The ribbon model of chain macromolecules is a useful tool for analyzing some of the large-scale shape features of these complex systems. Up to now, the ribbon model has been used mostly to produce graphical displays, which are usually analyzed by visual inspection. In this work we suggest a computational method for characterizing automatically, in a concise and algebraic fashion, some of the important shape features of these ribbon models. The procedure is based on a graph-theoretical and knot-theoretical characterization of three well-defined projections of a space curve associated with the ribbon. The labeled graphs can be characterized by the handedness of the crossovers in the ribbon that are the vertices of the graph. The method can be used to provide a fully algebraic representation of the changes occurring when a molecule, such as a protein, undergoes conformational rearrangements (folding), as well as to provide a shape comparison for a pair of related molecular ribbons. This algebraic representation is well suited for easy storage, retrieval, and computer manipulation of the information on the ribbon's shape. Illustrative examples of the method are provided.

Topological analysis of hydrogen bonding in protein structure

A recent study has shown that topological stereoisomers exist for the polypeptide chain in disulfide-containing proteins that are represented by non-planar graphs. This topological stereochemistry is demonstrated in the structure of variant 3 toxin in the venom of the North American scorpion Centruroides sculpturatus Ewing and the structure of toxin II from the North African scorpion Androctonus australis Hector. In this report, we found that a similar topological analysis can be applied to the hydrogen bonding in alpha-helices and beta-sheets within protein molecules, and we described the topological characteristics of chiral properties of protein secondary structure elements. Specifically, a closed right-handed alpha-helix of more than six residues long is shown formally to be non-planar and has the L topology. Antiparallel beta-sheets are planar. Two parallel beta-strands each of at least three residues in length, however, constitute a non-planar structural element and can have either L or D topology. The favored right-handed crossover for parallel beta-sheets has the L form, the same as the right-handed alpha-helix. This topological description of the hydrogen bonding in secondary structures may be extended to higher levels of protein structure and may provide a conceptual framework for studying complex protein architecture in general.

Biochemical topology: applications to DNA recombination and replication

Processes of DNA rearrangement such as recombination or replication frequently have as products different subsets of the limitless number of distinguishable catenanes or knots. The use of gel electrophoresis and electron microscopy for analysis of these topological isomers has made it possible to deduce physical and geometric features of DNA structure and reaction mechanisms that are otherwise experimentally inaccessible. Quantitative as well as qualitative characterization is possible for any pathway in which the fate of a circular DNA can be followed. The history, theory, and techniques are reviewed and illustrative examples from recent studies are presented.