Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 3.3, pp. 373-374   | 1 | 2 |

Section Stacked transformations

R. Diamonda*

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England
Correspondence e-mail: Stacked transformations

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A flexible molecule may require to be drawn in any of a number of conformations which are related to one another by, for example, rotations about single bonds, changes of bond angles or changes of bond lengths, all of which changes may be brought about by the application of suitable homogeneous transformations during the drawing of the molecule (Section[link]). With suitable organization, this may be done without necessarily altering the coordinates of the atoms in the coordinate list, only the transformations being manipulated during drawing.

The use of transformations in the manner shown below is straightforward for simply connected structures or structures containing only rigid rings. Flexible rings may be similarly handled provided that the matrices employed are consistent with the consequential constraints as described in Section[link], though this requirement may make real-time folding of flexible rings difficult.

Any simply connected structure may be organized as a tree with a node at each branch point and with an arbitrary number of sites of conformational change between one node and the next. We shall call such sites and their associated matrices `conformons'. The technique then depends on the stacking technique in which matrices are stored and later recovered in the reverse order of their storage.

One begins at some reference point deemed to be fixed in data space and at this point one stacks the prevailing viewing transformation. From this reference point one advances through the molecule along the structural tree and as each conformon is encountered its matrix is calculated. The product of the prevailing matrix with the conformon matrix is formed and stacked, and this product becomes the prevailing matrix. This product is constructed with the conformon matrix as a factor on the right, i.e. in data space as defined in Section[link], and is calculated using the coordinates of the molecule in their unmodified form, i.e. before any shape changes are brought about.

This progression leads eventually to an extremity of the tree. At this point drawing is commenced using the prevailing matrix and working backwards towards the fixed root, unstacking (or `popping') a matrix as each conformon is passed until a node is reached, which, in general, will occur only part way back to the root. On reaching such a node drawing is suspended and one advances along the newly found branch as before, stacking matrices, until another extremity is reached when drawing towards the root is resumed. This alternation of stacking matrices while moving away from the root and drawing and unstacking matrices while moving towards the root is continued until the whole tree is traversed.

This process is illustrated schematically in Fig.[link] for a simple tree with one node, numbered 1, and three conformons at a, b and c. One enters the tree with a current viewing transformation T and progresses upwards from the fixed lower extremity. When the conformon at a is encountered, T is stacked and the product [{\bi TM}_{a}] is formed. Continuing up the tree, at node 1 either branch may be chosen; we choose the left and, on reaching b, [{\bi TM}_{a}] is stacked and [{\bi TM}_{a}{\bi M}_{b}] is formed. On reaching the tip drawing down to b is done with this transformation, [{\bi TM}_{a}] is then unstacked and drawing continues with this matrix until node 1 is reached. The other branch is then followed to c whereupon [{\bi TM}_{a}] is again stacked and the product [{\bi TM}_{a}{\bi M}_{c}] is formed. From the tip down as far as c is drawn with this matrix, whereupon [{\bi TM}_{a}] is unstacked and drawing continues down to a, where T is unstacked before drawing the section nearest the root.


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Schematic representation of a simple branched-chain molecule with a stationary root and two extremities. The positions marked a, b and c are the loci of possible conformational change, here called conformons, and there is a single, numbered branch point.

With this organization the matrices associated with b and c are unaffected by changes in the conformation at a, notwithstanding the fact that changes at a alter the direction of the axis of rotation at b or c.

Two other approaches are also possible. One of these is to start at the tip of the left branch, replace the coordinates of atoms between b and the tip by [{\bi M}_{b}{\bf X}], and later replace all coordinates between the tip and a by [{\bi M}_{a}{\bf X}], with a similar treatment for the other branch. The advantage of this is that no storage is required for stacked matrices, but the disadvantage is that atoms near the tips of the tree have to be reprocessed for every conformon. It also modifies the stored coordinates, which may or may not be desirable.

The second alternative is to draw upwards from the root using T until a is reached, then using [{\bi TM}_{a}] until b is reached, then using [{\bi TM}'_{b}{\bi M}_{a}] to the tip, but in this formulation [{\bi M}'_{b}] must be based on the geometry that exists at b after the transformation [{\bi M}_{a}] has been applied to this region of the molecule, i.e. [{\bi M}'_{b}] is characteristic of the final conformation rather than the initial one.

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