Biology Reference
In-Depth Information
In contrast, DEE ensures the convergence to a globally optimal solu-
tion consistently. DEE operates on the systematic elimination of
rotamers that are not allowable to be parts of the sequence with the
lowest energy. The energy function in DEE is written in the form of
a sum of individual term (rotamer-template) and pairwise term
(rotamer-rotamer):
N
N
−
1
N
E
=
()
i
+
(, )
i
j
∑
∑
∑
(5)
r
r
s
i
=
1
i
=
1
ji
>
where
E
(
i
r
) is the rotamer-template energy for rotamer
i
r
of amino acid
i
,
E
(
i
r
,
j
s
) is the rotamer-rotamer energy of rotamer
i
r
and rotamer
j
s
of
amino acids
i
and
j
respectively, and
N
is the total number of residues
in the protein [31]. The pruning criterion in DEE is based on the con-
cept that if the pairwise energy between rotamer
i
r
and rotamer
j
s
is
higher than that between rotamers
i
t
and
j
s
for all
j
s
in a certain rotamer
set {
S
}, then
i
r
cannot be the global energy minimum conformation
(GMEC) and thus can be eliminated. Mathematically the idea can be
expressed as the following inequality [32]:
N
N
Ei
()
+
Ei
(, )
j
>
Ei
()
+
∑
Ei j
(, )
∀
{}
S
∑
(6)
r
r
s
t
t
s
ji
≠
ji
≠
So rotamer
i
r
can be pruned if the above holds true. Bounds implied
by (6) can be utilized to generate the following computationally more
tractable inequality [32]:
N
N
Ei
(
)
+
∑
min
Ei
(
,
j
)
>
Ei
(
)
+
∑
max
Ei
(
j
s
)
(7)
r
rs
t
t
s
s
ji
≠
ji
≠
The above inequality can be extended to eliminate pairs of rotamers.
This is done by determining a rotamer pair
i
r
and
j
s
which always con-
tributes higher energies than rotamer pair
i
u
and
j
v
for all possible rotamer
combinations. The analogous computationally tractable inequality is [32]:
N
N
ε
(, )
ij
+
min(, , )
ε
ijk
>
ε
( , )
i j
+
ma
xx( , , )
ε
ijk
(8)
∑
≠
∑
rs
rs t
uv
uv t
t
kij
≠
,
kij
,
t
where e is the total energies of rotamer pairs:
ε
(
i
,
j
)
=
Ei
(
)
+
E j
(
)
+
Ei
(
,
j
)
(9)
rs
r
s
rs
ε
(, , )
ijk
=
Eik
(, )
+
Ejk
( , )
(10)
rs t
r t
s t
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