data, most notably NMR spectroscopy. Other important appli- Distance Geometry: cations include enumerating the conformation spaces of small molecules, ligand docking and pharmacophore mapping in Theory, Algorithms, and drug design, and the homology modeling of protein structure. Chemical Applications 2 THEORY Timothy F. Havel Harvard Medical School, Boston, MA, USA One of the most significant developments in distance geo- metry 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 the- 1 Introduction 723 ory is certain to find manifold applications in computational 2 Theory 723 chemistry, not only in the analysis of simple geometric mod- 3 Algorithms 727 els of molecular structure, but also in more complete classical 4 Applications 735 and even quantum mechanical models. For these reasons we 5 Outlook 740 6 Related Articles 741 shall begin with a brief introduction to the geometric algebra 7 References 741 of a three-dimensional Euclidean vector space; a more detailed account, including its applications in classical mechanics, may be found in Ref. 2.
Abbreviation 2.1 Geometric Algebra RMSD D root-mean-square coordinate difference. Although its origins can be traced back over 150 years to the work of the German schoolmaster Hermann Grassmann,3 geometric algebra remains virtually unknown outside of a few 1 INTRODUCTION specialized branches of mathematics and theoretical physics (where it is more commonly known as Clifford algebra, Distance geometry is the mathematical basis for a geometric after one of its earliest developers). The three-dimensional theory of molecular conformation.1 This theory plays a role in Euclidean case is nevertheless not very difficult, and is all conformational analysis analogous to that played in statistical that is needed in order to begin to appreciate its utility. In mechanics by a hard-sphere fluid ... which can in fact be reading the following introduction, it should be kept in mind regarded as the distance geometry description of a mono- that geometric algebra simultaneously generalizes, and hence atomic fluid. More generally, a distance geometry description unifies, most of the algebraic structures with which the reader of a molecular system consists of a list of distance and chirality is probably already familiar, including the real and complex constraints. These are, respectively, lower and upper bounds numbers, vector algebra, and Hamilton’s quaternions. It is not on the distances between pairs of atoms, and the chirality of a substitute for linear algebra or differential calculus, but rather its rigid quadruples of atoms (i.e., R or S relative to some enriches them with new geometric content. given order). The distance geometry approach is predicated on the assumption that it is possible to adequately define the 2.1.1 The Rules of the Game set of all possible (i.e., significantly populated) conformations, or conformation space, of just about any nonrigid molecular The geometric product of vectors a, b,andcis an associa- system by means of such purely geometric constraints. By tive product, distributive with respect to vector addition, such that the square of any vector is the same as its length squared: Occam’s razor, we contend that any properties of the system that can be explained by such a simple model should be .ab/c D a.bc/, a.b C c/ D ab C ac, explained that way. 2 2 Distance geometry also plays an important role in the devel- .a C b/c D ac C bc, and a Djjajj .1/ opment of computational methods for analyzing distance geo- metry descriptions. The goal of these calculations is to deter- Note we have not required this multiplication to be commuta- tive, i.e., ab 6D ba in general. mine the global properties of the entire conformation space, One immediate consequence of this definition is that every as opposed to the local properties of its individual members. nonzero vector a has an inverse, namely This is done by deriving new geometric facts about the sys- tem from those given explicitly by the distance and chirality a 1 D a/a2 .2/ constraints, a process known more generally as geometric rea- soning. Although numerous constraints can be derived from Moreover, by the law of cosines, the usual inner product of knowledge of the molecular formula, in many cases (e.g., glob- vectors is ular proteins) additional noncovalent constraints are needed in ž 1 2 2 2 order to define precisely the accessible conformation space. a b D 2 .jjajj Cjjbjj jja bjj / These must be obtained from additional experiments, and thus D 1 .a 2 C b 2 .a b/ 2 / one of the best-known applications of distance geometry is the 2 1 determination of molecular conformation from experimental D 2 .ab C ba/.3/ 2 DISTANCE GEOMETRY: THEORY, ALGORITHMS, AND CHEMICAL APPLICATIONS
which is just the symmetric part of their geometric product ab. Together, these facts imply that abc abc is equal to the Thus it is natural to define the outer product of two vectors as trivector 2a ^ b ^ c. More generally, if we define the outer the antisymmetric part of their geometric product, i.e., product of a vector a with a bivector b ^ c to be the symmetric 1 part a ^ b D 2 .ab ba/.4/ 1 .a.b ^ c/ C .b ^ c/a/.9/ This outer product is clearly anticommutative, meaning a ^ 2 b D b^a, and vanishes if and only if a D ˛b for some of their geometric product, a straightforward calculation shows scalar ˛. The outer product a ^ b cannot be a vector, since that inversion in the origin (i.e., multiplying all vectors by 1) does not change it; neither can it be a scalar, since then a ^ .b ^ c/ D a ^ b ^ c D .a ^ b/ ^ c .10/ the geometric product ab D ažb C a ^ b would also be a scalar (which for linearly independent vectors a and c would Thus the outer product is also associative. contradict the above associativity condition). We can only It is also possible to define various ‘mixed’ products, which conclude that the outer product of two vectors is a new entity, are combinations of inner and outer products. Of particular called a bivector. interest is the inner product of a vector a with a bivector b ^ c, Given an orthonormal basis e1, e2 of e3 our vector space, which is defined as the antisymmetric part of their geometric we can expand any outer product using anticommutivity as product. In this case, a direct calculation yields the vector follows: ž 1 ž ž a .b ^ c/ D 2 .a.b ^ c/ .b ^ c/a/ D .a b/c .a c/b .11/ a ^ b D .a1e1 C a2e2 C a3e3/ ^ .b1e1 C b2e2 C b3e3/
D .a1b2 b1a2/e1e2 C .a3b1 b3a1/e3e1 Similarly, the inner product of two bivectors is defined as the symmetric part of their geometric product, in which case we C .a2b3 b2a3/e2e3 .5/ have:
This shows that every bivector can be expanded in terms of .a ^ b/ž.d ^ c/ D až.bž.d ^ c// the three elementary bivectors e1 ^ e2 D e1e2, e3 ^ e1 D e3e1, ^ D D .ažc/.bžd/ .ažd/.bžc/ and e2 e3 e2e3. Moreover, it is easily shown that ž ž 2 2 2 2 a cad .˛e1e2 C ˇe3e1 C e2e3/ D ˛ ˇ .6/ D det .12/ bžcbžd and hence these three bivectors must be linearly independent, Finally, the inner product of two trivectors is: so that the expansion is unique. This shows, in particular, that the space of bivectors is likewise three-dimensional. aždažeažf If we similarly define the outer product of three vectors as .a ^ b ^ c/ž.f ^ e ^ d/ D det bždbžebžf .13/ their antisymmetrized geometric product, i.e., cždcžecžf a ^ b ^ c D 1 .abc acb C cab cba C bca bac/.7/ 6 Determinants of matrices of inner products as above are a similar though longer calculation shows that a ^ b ^ c D commonly called Gramians. det.a, b, c/e1e2e3, where the determinant is of the matrix whose columns are the coordinates of a, b,andcversus the 2.1.2 Geometric Interpretation basis e1, e2,ande3. It follows that all trivectors are multiples of the trivector à D e1e2e3, which will henceforth be called The utility of the above, purely formal, algebraic system lies the unit pseudo-scalar. This has the interesting property of in the fact that all the quantities appearing in it have distinct behaving like the imaginary unit, since geometric meanings. This enables us to translate such basic geometric relations as congruence, perpendicularity, incidence, 2 2 à D .e1e2e3/ D .e1e2e3/.e3e2e1/ etc., into algebraic equations, and then to use the machinery 2 2 2 of geometric algebra to derive new relations. For example, D .e1e2/e3.e2e1/D e1e2e1 D e1 D 1 .8/ 2 2 In keeping with the fact that space is three-dimensional, all a D b , congruent, or outer products of four or more vectors must be zero. ažb D 0 , perpendicular .14/ Any element of the algebra, or multivector, can be uniquely expanded as a sum of a scalar, a vector, a bivector, and a This geometric interpretation also enables the entities that pseudo-scalar, for a total dimension of eight. The multilinearity appear in the algebra to represent many different physical of the geometric product shows that scalars and bivectors are quantities in very natural ways.2 unchanged on inversion in the origin, whereas vectors and Because most scientists are familiar with Gibbs’ vector trivectors change sign. This fact also implies that the product algebra, the easiest way to introduce this interpretation is to of an even number of vectors has no vector or trivector part, show how Gibbs’ algebra fits in to the more general geometric whereas odd products have no scalar or bivector part. This is algebra introduced above. Since the inner product is already described by saying that the product preserves parity. Another part of Gibbs’ vector algebra, we shall begin with the outer interesting transformation reverses the order of the vectors product of two vectors. Using the above formula for a ^ b in a product, e.g., abc D cba. It is easily seen that scalars together with the identity and vectors are unchanged by reversion, while bivectors and 2 2 trivectors change sign. Ãe1e2 D .e3e2e1/e1e2 D .e3e2/e1e2 D e3e2 D e3 .15/ DISTANCE GEOMETRY: THEORY, ALGORITHMS, AND CHEMICAL APPLICATIONS 3
2 (and similar identities involving Ãe3e1 and Ãe2e3), we find that D .sin.Âab/sin.Âbc/cos.//jjajjjjbjj jjcjj