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Constraining the Central Dihedral Angles in Acetylcholine

  tex2html_wrap3676 . Constraining the Central Dihedral Angles in Acetylcholine

As part of the investigation of Acetylcholine in Chapter gif it was necessary to allow the structure of the molecule to relax while constraining the two central dihedral angles tex2html_wrap_inline3564 and tex2html_wrap_inline3566 (See Figure gif). The constraints on the motion of the atoms necessary to achieve this are outlined in this appendix.

   figure2275
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 tex2html_wrap_inline3582 corresponds to the cis-planar arrangement of the bonds AB and CD. Thus, the angles between planes tex2html_wrap_inline3584 and tex2html_wrap_inline3586 and planes Y and tex2html_wrap_inline3590 defined in Figure gif must remain constant. In order to meet these conditions the following constraints can be applied to the motions of the ions:

  1. Fix tex2html_wrap_inline3592
  2. tex2html_wrap_inline3594 constrained to move along tex2html_wrap_inline3596 bond.
  3. tex2html_wrap_inline3598 constrained to move in plane Y.
  4. tex2html_wrap_inline3602 constrained to move in plane Z.
  5. tex2html_wrap_inline3606 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 tex2html_wrap_inline3598 may move out of the plane X. In order to ensure that D2 remains constant a correction must be applied to the position of tex2html_wrap_inline3606 , tex2html_wrap_inline3059 , tex2html_wrap_inline2954 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 gif shows a view along the tex2html_wrap_inline3626 bond in the direction indicated by unit vector tex2html_wrap_inline3628 . Let tex2html_wrap_inline3630 be the unit normal vector to plane Y and tex2html_wrap_inline3500 be the (non unit) vector between tex2html_wrap_inline3598 and tex2html_wrap_inline3606 . After a relaxation iteration tex2html_wrap_inline3630 will remain the same due to the constraints applied. However, tex2html_wrap_inline3500 and tex2html_wrap_inline3628 will change to tex2html_wrap_inline3646 and tex2html_wrap_inline3648 respectively. Finally we define

equation2306

   figure2315
Figure: A schematic diagram defining the vectors relating to the correction of dihedral angle D2 in acetylcholine.

In any permitted motion tex2html_wrap_inline3652 must remain constant, or equivalently

equation2322

must be constant.

After a relaxation, in general we will have tex2html_wrap_inline3646 , tex2html_wrap_inline3648 and tex2html_wrap_inline3658 such that

equation2331

Therefore a shift tex2html_wrap_inline3660 must be applied such that

  equation2343

Let tex2html_wrap_inline3662 , tex2html_wrap_inline3664 and tex2html_wrap_inline3666 which gives

  equation2361

from Equation gif. Furthermore, we wish to conserve the bond length tex2html_wrap_inline3606 - tex2html_wrap_inline3598 which gives

  equation2369

Solving Equations gif and gif simultaneously gives

eqnarray2376

which in turn allows us to calculate

eqnarray2384

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


next up previous contents
Next: References Up: No Title Previous: Orbital FT Kernels

Matthew Segall
Wed Sep 24 12:24:18 BST 1997