Virtual bond method

The earliest attempts at model analysis of polysaccharides -typified by the x-ray crystal structure analysis of amylose triacetate - were usually conducted in three steps ( L). In the first step, a model of the chain was established which was in agreement with the fiber repeat and the lattice geometry, as obtained from diffraction data. In the second step, the invariant chain model was packed into the unit cell, subject to constraints imposed by nonbonded contacts. This was followed, in the third step, by efforts to reconcile calculated and observed structure factor amplitudes. It was quickly realized that helical models of polysaccharide chains could be easily generated and varied using the virtual bond method. Figure 1 illustrates the generation of a two-fold helical model of a (l- U)-linked polysaccharide chain. 

Figure 1. Construction of a two-fold helical model of a polysaccharide with the virtual bond method. Increasing the length VB of the virtual bond is shown by...

Model Building and Refinement with the Virtual Bond Method... 

Figure 3. Two alternate methods of describing an a-D-glucose residue using the variable virtual bond method. The bond length, bond angle, and conformation angle for atom 05 are shown as ts, 6s, s-...

The model description and refinement based on the virtual bond method need not be restricted to a single monomer residue. 

Although various procedures are available for the model analysis of fibrous polymers, methods based on the virtual bond representation of the asymmetric residue may be of advantage in many cases. In the following, we describe one such method that began with simple procedures applied to polysaccharides, but has now been refined into a flexible and powerful model analysis tool that is simple to use with any class of polymer. Its use in the present case, however, is illustrated with examples drawn from the structure analysis of polysaccharides. 

The model of the residue can also be described by a second procedure, shown in Figure 3B. Two strings of atoms are used, beginning at separate ends of the virtual bond. The strings are not connected to one another, leaving two "open" bonds. This method is useful when the length of the virtual bond is to remain fixed during refinement. 

Since its introduction several years ago, the virtual bond, constrained optimization method has proved very useful in studies of polysaccharide crystal structure. Notable among the successes that can be ascribed to it are the structural determinations of the double-helical amylose (.11), the cellulose polymorphs of different chain polarities (.12, 13), and of a number of other polysaccharides and their derivatives. As described in a review of amylose structures elsewhere in this volume, the use of this refinement method has produced structural detail that has previously been unavailable (ll). These results have provided much-needed... 

The flexible helix modeled here is best described by the entire array of conformations it can assume. A comprehensive picture of this array is provided by the three-dimensional spatial probability density function Wn(r) of all possible end-to-end vectors (25, 35). This function is equal to the probability per unit volume in space that the flexible chain terminates at vector position relative to the chain origin 0,as reference. An approximate picture of this distribution function is provided by the three flexible single-stranded B-DNA chains of 128 residues in Figure 5(a). The conformations of these molecules are chosen at random by Monte Carlo methods (35, 36) from the conformations accessible to the duplex model. The three molecules are drawn in a common coordinate system defined by the initial virtual bond of each strand. For clarity, the sugar and base moieties are omitted and the segments are represented by the virtual bonds connecting successive phosphorus atoms. 

Figure 5. Computer generated perspective representation of single-stranded B-DNA chains. The 128-residue chains are represented by the sequence of virtual bonds connecting successive phosphorus atoms, (a) Three representative flexible helices generated by Monte Carlo methods (b) the regular

As a further example demonstrating the application of the overlap determinant method to cycloadditions, let us analyze the 2 + 2 dimerization of ethene. As has already been stated above, the irreducible core of this reaction for the ground state process is formed by the set of two disappearing ethene 7t bonds and the set of two newly created cyclobutane ct bonds. The fact that we are interested here in the photochemical process finds its reflection in the formal replacement of one of the ethene n bonds by the corresponding virtual bond. Let it be for example bond 7134. The bonds forming the irreducible core of photoreaction are then again described by the usual linear combinations (57) ... 

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