Biology Reference
In-Depth Information
The corresponding Gauss function is given by the following equation:
(
)
(
)
(
)
⎛
2
⎞
⎛
2
⎞
⎛
2
⎞
−−
xx
−−
yy
−−
zz
1
i
i
i
t
i
H
=
exp
exp
exp
⎜
⎟
⎜
⎟
⎜
⎟
t
sum
2
2
2
H
2
s
2
σ
2
σ
⎝
⎠
⎝
⎠
⎝
⎠
x
y
z
where
x
yz
(the position of
the effective atom for the
i-th
residue), while parameters
,,
t
H
denotes the hydrophobicity density at coordinate
s
,,
i
i
i
xyz
denote the coordi-
nates of the central point of the ellipsoid (treated as the geometric center of the
entire molecule) whose size is determined by values
σσσ
x
,
,
z
(calculated as 1/3 of
y
the greatest distance along each axis). The calculation of
x y z
follows a prede-
termined orientation of the molecule, with its geometric center located at the origin
of the coordinate system. Inter-atomic pairwise distances are calculated for the
whole molecule. The greatest distance determines the orientation of the molecule
along the X axis, while the greatest distance between two projections of atom positions
on the YZ plane determines its orientation along the Y axis. Values
σσσ
,
,
σσσ
,
,
x y z
can be
calculated for the orientation described above. Their sum is used as a normalizing
factor for the distribution.
The value of the presented function at coordinates
x, y, z
is interpreted as the
corresponding hydrophobicity density.
In order to ensure uniformity of the presented model we need to determine the
preferred spatial orientation of the protein molecule. The corresponding procedure
is as follows:
1. Calculate the coordinates of the geometric center of the molecule;
2. Shift the molecule in such way that its geometric center coincides with the origin
of the coordinate system;
3. Following the transformation applied in step 2, calculate the greatest pairwise
distance between two atoms belonging to the molecule;
4. The atoms identified in step 3 determine the spatial orientation of the molecule -
the line which connects them should be parallel to (or coincide with) the X axis
of the coordinate system;
5. Given the new orientation of the protein molecule, project the positions of its
atoms onto the YZ plane and locate two atoms for which the pair-wise distance
between the corresponding projections is greatest;
6. The line connecting the two atoms identified in step 5 should run parallel to (or
coincide with) the Y axis of the coordinate system. In order to achieve this, rotate
the molecule about the X axis, as required.
7. Given the new orientation of the molecule, locate atoms which are separated by
the greatest distance from the center of the coordinate system along each axis
(two atoms for each axis);
8. Increase the distances determined in step 7 by 9 Å (the cutoff distance for hydro-
phobic interactions) in each direction;
9. Divide the distances calculated in step 8 by 6, thereby deriving values for
σσσ
x
,
,
z
(in accordance with the three-sigma rule).
y
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