. Constraining the Central Dihedral Angles in Acetylcholine

As part of the investigation of Acetylcholine in Chapter it was necessary to allow the structure of the molecule to relax while constraining the two central dihedral angles and (See Figure ). The constraints on the motion of the atoms necessary to achieve this are outlined in this appendix.

**Figure:** A
schematic diagram defining the planes relating to the constraints on
the dihedral angles *D*2 and *D*3. Plane *X* is shown in red, plane
*Y* in green and plane *Z* in blue. Oxygen atoms are shown in red,
nitrogen in light blue and carbon in grey, hydrogen atoms are omitted
for clarity.

The dihedral angle defined by four bonded atoms A-B-C-D is defined by
the angle between the planes ABC and BCD viewed from the side of A,
such that a positive angle is given by a clockwise rotation of the far
end with respect to the near and an angle of corresponds to
the *cis*-planar arrangement of the bonds AB and CD. Thus, the
angles between planes and and
planes *Y* and defined in Figure
must remain constant. In order to meet these conditions the following
constraints can be applied to the motions of the ions:

- Fix
- constrained to move along bond.
- constrained to move in plane
*Y*. - constrained to move in plane
*Z*. - constrained to move in plane
*X*.

Figure shows a view along the bond
in the direction indicated by unit vector . Let be the unit normal vector to plane *Y* and be the (non
unit) vector between and . After a relaxation
iteration will remain the same due to the constraints
applied. However, and will change to
and respectively. Finally we define

**Figure:** A schematic diagram defining the vectors relating to the
correction of dihedral angle *D*2 in acetylcholine.

In any permitted motion must remain constant, or equivalently

must be constant.

After a relaxation, in general we will have , and such that

Therefore a shift must be applied such that

Let , and which gives

from Equation . Furthermore, we wish to conserve the bond length - which gives

Solving Equations and simultaneously gives

which in turn allows us to calculate

The smallest shift obeying these relations is chosen and applied to correct the deviation from the constraint.

Wed Sep 24 12:24:18 BST 1997