Constraints chiral

Fig. 9.76 The stereochemistry about tetrahedral atoms can be maintained with an appnrpriate chiral constraint.

The distance matrix of upper and lower bounds now describes the complete conformation space of the molecule but unfortunately cannot describe its chirality, since the distance matrix is invariant with respect to chirality. Chiral constraints are added to supply this missing information a chiral constraint is specified as the signed volume of the tetrahedron formed by the four substituents attached to a chiral center. The volume is calculated as a vector... 

All the distance and chiral constraints can be assigned automatically by the distance geometry program directly from connectivity, atom type, bond length, and bond angle data (132). Additional distance and chiral constraints can also be assigned by the user to generate specific conformations or intermolecular interactions (several molecules at a time can also be entered into a distance matrix). 

Distances for each atom pair are randomly chosen between their lower and upper bounds. These distances are then converted into three-dimensional coordinates and refined against a simple error function made up of contributions from upper and lower bound violations and chiral constraint violations to ensure that the structure meets all distance and chiral constraints. The details of converting the distance matrix to three-dimensional coordinates are beyond the scope of this chapter but are provided in Crippen s text (126) and in an upcoming review article (133). 

Figure 15 Three sets of chiral constraints are required to ensure planarity of a double bonded system.

Chiral constraints also can be used on united atom models that do not include hydrogens in this case, the chiral constraint includes three nonhydrogen substituents and the asymmetric carbon itself. For structures whose absolute configuration is unknown, chirality can be sampled randomly by simply omitting the chiral constraint at asymmetric carbons of unknown configuration. 3... 

Chiral constraints are also used to maintain planarity by constraining four atoms to have zero volume. Because the chiral volume is more sensitive than the distance error term to minor variations from planarity, it is used to constrain planar groups more accurately. 

A minimum of three chiral constraints are used to maintain planarity at unsaturated bonds (aromatic, amide, ester, etc.) atoms 1-2-3-4, 1-5-6-3, and... 

Figure 7 Chiral constraint expressed as the volume V of the tetrahedron enclosing four atoms and the chiral error funaion, fchirai-...

Z-6-5-4 in Figure 8. Amide and ester torsions can be allowed to twist up to about 10° by allowing a small deviation (e.g., O.lA ) on the 1-2-3-4 constraint in Figure 8. An amide torsion can be randomly sampled between cis and trans by assigning 1-4 distance bounds as if it were a freely rotatable bond (for distances 1-2, 1-4, and 2-3 in Fig. 8), which will randomly sample all torsional angles, and assigning the three chiral constraints (1-2-3-4, 1-5-6-3, and... 

A chiral constraint plus a distance constraint also can be used to constrain nonplanar torsional angles. The 1-4 distance constraint cannot define a torsional angle by itself for example, 90° and —90° have the same 1-4 distance, but their tetrahedral volumes are equal in magnitude but opposite in sign. 

Struaures generated by distance geometry will not satisfy all distance and chiral constraints perfectly. Minor violations less than 0.5 A generally do not... 

During the optimisation of the structure against the distance constraints it is usual to incorporate chiral consiraints. These are used to ensure that the final conformation is the desired stereoisomer. Chiral constraints are necessary because the interatomic distances in two enantiomeric conformations are identical and as a consequence the wrong isomer may quite legitimately be generated. Chiral constraints are usually incorporated into the error function as a chiral volume, calculated as a scalar triple product. For example, to maintain the correct stereochemistry about the tetrahedral atom number 4 in Figure 9,16, the following scalar triple product must be positive ... 

Fig 916- The stereochemistry about tetrahedral atoms can he maintained with an appropriate chiral constraint... 

Fig. 917 A double bond can be forced to adopt a planar conformation through die use of appropriate chiral constraints.

The other stereoisomer corresponds to a negative chiral volume. Chiral constraints are included in the penalty function by adding terms of the following form ... 

Many enhancements have been made to the basic distance geometry method. Some of the most useful enhancements result from the incorporation of chemical information. For example, if the lower bound for the 1,4 distances is set to a value equivalent to a torsion angle of 60° rather than one of 0° then eclipsed conformations can be avoided. Similarly, amide bonds can be forced to adopt a nearly planar structure by an appropriate choice of distance bounds and chiral constraints. 

While one could require these conformations to satisfy a wide variety of geometric conditions by means of constraints on suitable polynomials in the interatomic squared distances and signed volumes, it turns out that the simplest possible such constraints are also the most widely useful. These are lower and upper bounds on the interatomic squared distances themselves, together with the signs (+1, —1, or 0) of the volumes of selected quadruples of atoms. The latter, called chirality constraints, determine the chirality of the quadruple, or force it to be planar if the sign is zero. The totality of constraints of this form is called a distance geometry description. Experience has shown that most conformation spaces of practical interest in chemical problems can be accurately described by means of these simple constraints alone. 

Optimizing these coordinates versus an error function which measures the total violation of the distance and chirality constraints, usually by some form of simulated annealing. 

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