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Untangling Molecular Knots

Buks van Rensburg

When Topology Meets Chemistry: A Topological Look at Molecular Chirality. Erica Flapan. xiv + 241 pp. Mathematical Association of America and Cambridge University Press, 2000. $69.95 cloth, $24.95 paper.

Open any chemistry book, and you will notice neatly drawn diagrams representing molecules of various kinds. These diagrams are often planar (they can be drawn on a flat sheet of paper without two lines crossing one another), but chemists are keenly aware that they are unrealistic representations and that the three-dimensional shapes of molecules are critical in determining their functionality.

Classical examples of this are the optically active compounds first observed by Pasteur. An optically pure substance rotates in either a left- or a right-handed way a beam of polarized light that shines along a path through the substance. The molecules responsible for optical activity come in two varieties, one left-handed and the other right-handed, and they are mirror images of one another. Such molecules are called chiral, and their existence exemplifies the importance of three-dimensional structure in molecules.

Two molecules that have the same chemical formula but are not identical are called isomers. Structural isomers are isomers that have different molecular bond graphs. Examples are butane and isobutane, two forms of a hydrocarbon that contains four carbon atoms: In butane the four carbon atoms are arranged in a row, whereas in isobutane three of the carbon atoms are bonded to a central carbon atom, giving a branched structure. Rigid stereoisomers are molecules that have the same molecular bond graphs but cannot be rotated as rigid structures to become identical. Chiral pairs of optically active compounds are often rigid stereoisomers: One species is the mirror image of the other, and if the rigidity is ignored, it can be deformed into the other species. Topological stereoisomers are molecules that have identical abstract molecular graphs but cannot be deformed into one another (even if they are considered to be made of putty or rubber) without tearing or breaking and reconnecting some part of the molecule.

In When Topology Meets Chemistry, Erica Flapan considers chirality and topological stereoisomers in chemistry. A careful, easy-to-read introduction prepares the reader well for the material that is presented in later chapters. Particular attention is paid to stereoisomers, and the notion of chirality is discussed in an intuitive manner. For example, Flapan notes that chemists, because they are accustomed to working with rigid molecules, often use the following definition: "A molecular bond graph is achiral if it can be rigidly rotated to its mirror image. Otherwise it is chiral." Chirality is explored with the idea of symmetry presentations: A molecule is in a symmetry presentation if it is already its own mirror image, or if it can be rotated to its mirror image. The idea of a symmetry presentation leads naturally to the notion of "rubber gloves." If a molecule can change into its mirror image but has no chemically accessible symmetry presentation, then it is called a Euclidean rubber glove. As for many other concepts in this book, a very intuitive and sensible explanation is given for this: Think of two gloves, right and left. By turning the right glove inside out, the left is obtained, but there is nowhere a symmetry presentation of the glove.

This example is typical of Flapan's efficient and intuitive presentation of complicated ideas in topology. She mixes ideas from topology well with examples and describes effectively the reasons why topology is relevant. For example, the chirality of the Möbius ladder is discussed in chapter 3, and after significant arguments it is shown that a Möbius ladder with an odd number of rungs is topologically chiral. Flapan immediately points out the relevance of this theorem by noting that there exist various types of molecules with molecular bond graphs related to the graph of a Möbius ladder. One such molecule is triple-layered naphthalenophane, which she then proves to be intrinsically chiral.

There are other beautiful applications of the Möbius ladder in this book. For example, [m][n]paracyclophane is a molecule that is chemically chiral but has topologically achiral molecular graphs. The chirality of this molecule is understood if its molecular cell complexes (obtained by replacing impenetrable ring complexes such as benzene by an edge) are topologically intrinsically chiral (that is, if its chirality is a property of the molecule itself and not of its embedding in space). The proof of this, found on pages 131 and 132, proceeds by showing that the molecular cell complex is homeomorphic to a Möbius ladder with three rungs.

After considering chirality, rubber gloves, link polynomials and Möbius ladders in the first four chapters, Flapan considers the topology and symmetry of graphs in chapters 5 and 6. The work of John Horton Conway and Cameron McA. Gordon is reviewed, and the intrinsic chirality of complete graphs is discussed. Rigid and nonrigid symmetries of graphs are considered in chapter 6. Chapter 7 focuses on the topology of DNA. Supercoiling, writhing and tangle calculus are discussed. Flapan also shows how Claus Ernst and DeWitt Sumners used topological arguments building on the theory of tangles to analyze the action of the enzyme TN3 Resolvase on DNA molecules.

Well-written, well-organized and a pleasure to read, this book is full of interesting results, illustrated with line diagrams wherever needed. Every mathematician or chemist interested in the notions of chirality and symmetry should have a copy within easy reach.—Buks van Rensburg, Mathematics and Statistics, York University, Ontario, Canada

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