Distance Geometry

Distance Geometry

Read here about the relationship of tensegrity to distance geometry theories, possible a significant step towards comprehending the role of tensegrity in molecular geometry.

Overview

Timothy Havel wrote:

Distance geometry is the mathematical basis for a geometric theory of molecular conformation. This theory plays a role in conformational analysis analogous to that played in statistical mechanics by a hard-sphere fluid... which can in fact be regarded as the distance geometry description of a monoatomic fluid. More generally, a distance geometry description of a molecular system consists of a list of distance and chirality constraints. These are, respectively, lower and upper bounds on the distances between pairs of atoms, and the chirality of its rigid quadruples of atoms (i.e., R or S relative to some given order). The distance geometry approach is predicated on the assumption that it is possible to adequately define the set of all possible (i.e., significantly populated) conformations, or conformation space, of just about any nonrigid molecular system by means of such purely geometric constraints. By Occam’s razor, we contend that any properties of the system that can be explained by such a simple model should be explained that way.

Distance geometry also plays an important role in the development of computational methods for analyzing distance geometry descriptions. The goal of these calculations is to determine the global properties of the entire conformation space, as opposed to the local properties of its individual members. This is done by deriving new geometric facts about the system from those given explicitly by the distance and chirality constraints, a process known more generally as geometric reasoning. Although numerous constraints can be derived from knowledge of the molecular formula, in many cases (e.g., globular proteins) additional noncovalent constraints are needed in order to define precisely the accessible conformation space. These must be obtained from additional experiments, and thus one of the best-known applications of distance geometry is the determination of molecular conformation from experimental data, most notably NMR spectroscopy. Other important applications include enumerating the conformation spaces of small molecules, ligand docking and pharmacophore mapping in drug design, and the homology modeling of protein structure.

One of the most significant developments in distance geometry over the last few years has been the realization that the underlying theory is actually a special case of a more general theory, known as geometric algebra. This more general theory is certain to find manifold applications in computational chemistry, not only in the analysis of simple geometric models of molecular structure, but also in more complete classical and even quantum mechanical models. [1]

Globally rigid tensegrities... represent the 'boundaries' of the conformation space (i.e. internal configuration space) of a system of N points in a Euclidean space of arbitrarily high dimension. For this reason, and because the forces among a system of particles can be viewed as a stress in a tensegrity framework, tensegrtiy theory would seem to have a great deal to say about the classical N-body problem in mechanics.... these ideas might profitably be generalized to quantum N-body problems as well. [2]

Links and References

[1] Distance Geometry: Theory, Algorithms, and Chemical Applications by Timothy F. Havel, Harvard Medical School, Boston, MA, USA [2] The Role of Tensegrity in Distance Geometry by Havel, in Rigidity Theory and Applications