The most common way of determining the structures of molecules in a solution is nuclear magnetic resonance (NMR). The signals measured are caused by the flipping of the spins of paramagnetic nuclei in a magnetic field. The strengths of the signals depend on the proximity of the nuclei to each other. The interactions are short range, within 0.5 nanometers (nm). NMR is used as a sensitive tool to investigate how close atoms are to each other, which in turn provides information about the angles, especially dihedral angles, whose twists can change the molecule from one conformation to another. The most commonly used paramagnetic isotope is the simple hydrogen-1 isotope that has two spin states--up and down. It is plentiful in the functional groups of biologically interesting molecules.

The problem chemists face in revealing the structure of a molecule by NMR is precisely the problem that the study of distance geometry concerns itself. The chemist knows some of the interatomic distances because of the fixed geometries of many substructures in the molecule. However, there are bonds that are free to turn on dihedral angles whose values could either bring the atoms close to one another or far away from each other as the molecules try to achieve geometries of lowest energy. These are the unknown distances that are sought. If they are found from analyzing NMR signal strengths, they will fix the values of the dihedral angles that characterize the molecule’s geometry.

This very slim book (54 pages, including references) is based on the lecture notes for a course the authors developed on distance geometry. Each chapter is divided into several short sections of a few pages in length. Each section has a few exercises for the student to solve in order to set up the discussion in the subsequent section. There are seven chapters in this book. The first chapter is an introduction that presents the definitions, nomenclature, and symbols (mostly from graph theory) used throughout the book. The second chapter is a discussion of the distance geometry problem--what it is and the difficulty and complexity of finding solutions. The third chapter relates the distance geometry problem to optimization. Chapters 4 through 6 are related in the continuous development of the distance geometry problem in molecular structure determination. The seventh chapter is a short conclusion to the book that mentions briefly the difficulties of uncertainty in measured NMR signals and their impact on solving the distance geometry problem. This is a very serious problem that can lead to the prediction of unrealistic geometries.

The mathematically inclined chemist could relate best to this book. The authors provide a graph theoretical basis for what chemists do empirically in the laboratory when they elucidate the structure of a large molecule. The chemist knows which distances are fixed and will serve as reference points for the analysis of other signals that will lead to the molecular geometry. Chemists will then simulate signal strengths and relate them to the distances that generate them. This process is iterative until an acceptable, self-consistent assignment of distances is obtained. The authors emphasize the importance of the order of the labeling of vertices (the atoms in the molecule). This, too, is a problem that chemists face when they set up the geometry of the molecule for computation in creating the Z-matrix of bond distances, angles, and dihedral angles that can be modified computationally with the greatest ease.

I have worked in this area of research myself and appreciate the clarity and thoroughness of the presentation of this topic. I recommend it to any computational chemist, NMR spectroscopist, or biochemist who is working on determining molecular structure.