Homochirality classes

While a sense of chirality can be defined arbitrarily for the members of each individual heterochiral pair in such a set, so that one member within each pair is right-handed and the other left-handed, it does not follow from this that the right-handed member of one pair bears a well-defined relationship to the right-handed member of another pair. In other words, it is not, in general, possible to partition the members of the set C U C into two homochirality classes, right-handed (R) or left-handed (L). This is the homochirality problem of the title. 

FIGURE 1 Unlabeled triangles in the plane. Enantiomorphs are characterized by the orientation, clockwise vs anticlockwise, of the sides arranged in the order largest (/) medium (m) smallest (s). R and L spaces are separated by a subspace of achiral triangles. 

L spaces of Fig. 1 constitute homochirality classes because enantiomor-phous triangles cannot be chirally connected passage between the two spaces is impossible without crossing an achiral boundary. 

The homochirality problem arises whenever there exists a pathway of continuous deformation that connects two enantiomorphous objects but does not require passage through an achiral point. Under these circumstances, it becomes meaningless to speak of such objects as right-handed or left-handed. The case of the asymmetric tetrahedron is the simplest example. 

The asymmetric tetrahedron with unlabeled vertices is the three-dimensional analog of the unlabeled scalene triangle. While conversion of such a tetrahedron to its enantiomorph by way of an achiral tetrahedron is certainly not excluded, as illustrated in Fig. 2 for interconversion by way of a Cj-symmetric intermediate, achiral pathways are easily circumvented because the set of achiral unlabeled tetrahedra in E, unlike the set of achiral unlabeled triangles in E, does not form a boundary between heterochiral sets. 

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The absence of an achiral boundary along the conformational enantiomerization trajectory of chemically achiral compounds, such as the ones discussed above, precludes partitioning of conformations along the path into homochirality classes. As noted above, under such circumstances it becomes meaningless to speak of these conformations as right-handed or left-handed because no point can be defined, other than arbitrarily, where right switches to left and vice versa. 

The unlabeled triangle is the simplex in E (2-simplex) and the unlabeled tetrahedron is the simplex in (3-simplex) evidently, whether enantiomorphous -simplexes can be partitioned into homochirality classes depends on the dimension of E". Recall that an /j-simplex is a convex hull of + 1 points that do not lie in any (n - l)-dimensional subspace and that are linearly independent that is, whenever one of the points is fked, the n vectors that link it to the other n points form a basis for an n-dimensional Euclidean space An n-simplex may be visualized as an n-dimensional polytope (a geometrical figure in E" bounded by lines, planes, or hyperplanes) that has n + vertices, n n + )/2 edges, and is bounded by n + 1 (u — l)-dimensional subspaces. It has been shown that the homochirality problem for the simplex in E is shared by all -sim-... 

Fig. 12 represent particular stationary points on the multidimensional potential energy surface and are chirally connected through a continuous reaction pathway that involves no achiral intermediates this is evidently the case because the product of the reaction sequence is not racemic. Hence, they cannot be partitioned into R and L homochirality classes. 

Certain types of knots and links exist as topologically chiral enan-tiomorphs. Such enantiomorphs cannot be interconverted by continuous deformation ( ambient isotopy ). Homochirality classes can therefore be defined for this type of mathematical object. ... 

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