Biomedical Engineering Reference
In-Depth Information
Different LJ potentials are used because the native contact distances r
i
0
can be much
longer in the C
˛
model. The contacts are coarse-grained to be between the C
˛
atoms,
which can be as distant as 14 A. The r
6
is much broader than the r
10
and can lead
to unphysical structures in unfolded states as native pairs interact at long distances.
The LJ potentials are well tested and work for many systems, but they have
limitations for certain applications because the LJ potential has an excluded volume
that moves with the minimum. The effective size of two atoms in contact grows
with r
i
0
. This additional excluded volume has little effect on the entropy of unfolded
conformations since mostly noncontacts are interacting, but has a large effect on the
entropy of the folded ensemble where most contacts are formed. In cases where the
user wants to control the excluded volume term [
32
,
43
], an attractive Gaussian well
coupled with a fixed hard wall-excluded volume is used,
12
!
1
C
G.r
ij
;r
i
0
/
NC
r
ij
C
G
.r
ij
;r
i
0
/
D
1
C
1;
(7)
where
exp
h
0
/
2
=.2
2
/
i
:
G.r
ij
;r
ij
r
ij
0
/
D
.r
ij
(8)
This unusual construction anchors the depth of the minimum at -1. The width of
the Gaussian well is determined to model the variable width of the LJ potential.
C
AA
.1:2r
i
0
;r
i
0
/
1=2 so is defined such that G.1:2r
i
0
;r
i
0
/
D
1=2 giving
.r
i
0
/
2
=.50 ln 2/.If
NC
is significantly smaller than r
i
0
2
D
,(
7
) reduces to the
more pedagogical form,
NC
r
ij
12
C
G
.r
ij
;r
i
0
/
!
C
G.r
ij
;r
i
0
/
for
NC
r
i
0
:
(9)
The flexibility of the Gaussian approach also allows for multiple basin contact
potentials for energy landscapes with multiple minima (see Sect.
4.3
). Using
multiple LJ potentials with different locations of the minima is not a viable option
because the longest LJ potential would occlude the others with its excluded volume
term. A multibasin Gaussian potential C
MB
for minima taken from two structures
E
x
˛
and
x
ˇ
is given by [
32
],
E
12
!
NC
r
ij
1
C
G.r
ij
;r
i
˛
/
1
C
G.r
ij
;r
ij
ˇ
/
C
MB
.r
ij
;r
i
˛
;r
ij
ˇ
/
D
1
C
1:
(10)
Analogous to (
7
), this construction fixes the depth of both minima at
1.
All of the various potential shapes are presented in Fig.
3
. It should be noted that
the folding temperature (defined in Sect.
4.1.1
) is typically 0.2-0.3 reduced units
higher for the Gaussian potential as compared to LJ because the extra excluded
volume in the LJ potential destabilizes the native state.
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