Constraining the Central Dihedral Angles in Acetylcholine

  . 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 D2 and D3. 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:

1. Fix
2. constrained to move along bond.
3. constrained to move in plane Y.
4. constrained to move in plane Z.
5. constrained to move in plane X.

This removes 8 degrees of freedom form the motion of the molecule as it also prohibits bulk rotational and translation motions. These constraints fix the angle D3, however a change in D2 remains possible as the ion may move out of the plane X. In order to ensure that D2 remains constant a correction must be applied to the position of , , and associated hydrogen ions. As the movement of the ions in one iteration will be small, a linear shift may be applied to correct any such deviation.

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 D2 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.