T h e re g u la r o b ject p ro d u ce d by tesse lla tio n m u st now be
m apped onto each protein and used to produce an unbiased sample
of the m olecular surface. The centres of mass of the m olecule to be
sliced and the tessellated icosahedron were superim posed. The c o
ordinates of the tessellated icosahedron were then projected onto the
p ro te in su rfa ce . T h e e n se m b le was then ro ta te d so th a t each
tessellation point in turn lay uppermost on the Z-axis. A slice of the
protein surface was taken, centred upon this point. This slice was a
64x64 array of surface heights taken at intervals, hence a 32Â x
32Â area was represented. The van der W aals surface for the protein
was then calculated using a standard set of radii (Table 2.3). The
grea test surface h e ig h t found for each su rface e le m e n t was then
f o u n d .
Once the height of each surface element, z'(i,j), was known the
data was reduced to a set of integers in the range 0 to 63, intervals
of were taken and the scale arranged so that the highest surface
p o in t was denoted as 63 and anything further than 16Â below this
p o in t was denoted as 0. This method o f quantizing the surface into
d is c r e te e le m e n ts m in im ise d the im p a c t o f sm all v a ria tio n s in
c o n f o r m a ti o n .
%
Figure 2.5
The 372 vertices of a tessellated icosahedron projected onto a sphere. The object was created by the tessellation corresponding to h=4, k=3.
Group Type van der Waals radius (Â) backbone N 1.7 backbone C a 2 . 0 backbone C a (proline) 1 . 8 backbone C 1.7 backbone 0 1.4 sidechain CH CH2 CH3 2 . 0 N 1.7 NH 1 . 8 NH2 2 . 0 SH 1.5 S 1 . 8 hydroxyl oxygen 1 . 6 carboxylic oxygen 1 . 6 amide carbon 1.7 carboxylic carbon 1.7 Table 2.3
The van der Waals radii used for each atom type. These radii are taken from Gellately & Finney (1982).
To reduce further these effects, the m aps were sm oothed by adding
in h e ig h t c o m p o n e n ts from the n e a re s t n e ig h b o u r and d iag o n a l
elements, so that for a particular element i j
z ( .v ) =
T
z ' ( i - l j ) + z ' ( i + l j ) + z ' ( f j - l ) + z ' ( i . / + l )
12
z ' ( M . / - l ) + z '( » + l . / - l ) + z ' ( i - l . / + l ) + z ' ( i + l . / + l ) 2 4
T h e fra c tio n a l addition e n su red th at th e ra n g e o f h eig h ts
rem ained the same. Finally, to enable a one-to-one co rresp o n d en ce
between surface elements of host and guest maps, the surface of the
antigen had to be inverted in the x-axis (n o tio n ally turning the
surface upside down).
This procedure was carried out for both the host and guest
m olecules. The num ber of slices taken of the host m olecule was
g en erally restric te d to a know n rec ep to r reg io n (see the resu lts
Chapters for further details). The resulting data files were stored in
D A P form at using low level I/O functions so that they could be
quickly retrieved when required.
2.8. G lo b a l s e a r c h
A global search of the docking problem involves exploring six
degrees of freedom (Figure 2.6). Four rotational degrees of freedom
were accounted for by taking slices of the host and guest molecules.
An a d d itio n al degree o f freed o m was the p e rp e n d ic u la r d istan c e
betw een the slices, r. The slices were held at a distance such that
th eir surfaces ju s t touched, the potential was calculated, and then
A N T I G E N MAP A N T I B O D Y MAP dx 0 F i g u r e 2.6
The degrees o f freedom exam ined during the docking process (c.f. Figure 1.15). The guest m olecule is shown upperm ost with the h o st m o le cu le below . The a n tib o dy and an tig en m aps show the n otional d ocking area. The rotational degrees of freed o m 0 and (j)
bring any section of the guest m olecule into this docking area. The an g les a and p are the e q u iv alen ts for the ho st m o lecu le. The p e rp e n d icu lar d istance betw een the two m o lecu lar surfaces, r , can then be varied. A rotation to p e rp e n d icu lar to the plan e o f the su rfa c e s c o m p le te s the c la ss ic six d e g re e s o f fre e d o m . T w o p erp en d icu lar shifts in the plane of the docking area, d x and d y , w e re used to in c re a s e c o v e ra g e of the search sp a ce w ith o u t significantly increasing com putational costs.
1 J
they were then moved together in ^ A steps until there two surface
ele m e n ts o v e rla p p ed by 5Â (F ig u re 2.7). T his level o f o v erlap
re p re se n ts an u n a c c e p ta b le clash b e tw ee n the d o c k in g su rfaces.
P re cise atom ic d etail was lo st during the co d in g and sm oothing process. This meant that the rotation in the plane of the slice, o , the
final degree of rotational freedom , could not be directly carried out
on the slice. Instead, the protein structure was rotated in 8° steps
and 45 maps slices taken for each host surface segment.
This gives six deg rees o f freedom . H o w ev e r, in the DA P
a rc h ite c tu re the tra n sla tio n o f the su rfaces c o u ld be done very
quickly, since neighbouring processor are directly connected. It was
therefore decided to m ake small shifts in the plane o f the slices,
d x ,d y . T h is in c re a s e d the c o v e ra g e o f each m o le c u le w ith o u t
significantly decreasing the speed of the algorithm.
Only the binding region of the host m olecule was used in the
study, hence only 64 maps were taken instead of the global 432. The
n u m b e r o f o rie n ta tio n s c o n sid e re d , allo w in g for all d e g re e s of
freedom , was
N orientations = 432 guest molecule maps
X 64 host molecule maps
X 45 rotational
X 5 internal dx shifts X 5 internal dy shifts X 1 0 height param eters