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To
the previous RGA-page
Protein Folding: Lennard-Jones Clusters
The problem
The problem is taken from physical chemistry:
find such a structure of a protein, a cluster of N atoms,
interacting via the Lennard-Jones (dimensionless non-quantizied
pair ) potential
:
V
( r ) = r-
12
- 2 r- 6 ,
that its energy E is (globally) minimal.
There are fairly smart approaches to solving the problem of
LJ-
clusters, which exploit the known analytical form of the objective
function (OF : potential energy
of the microcluster conformation).
Here we would like just to illustrate the capabilities
of the RGA-algorithm, i.e. to solve the problem with no use a priori information on
the OF by as few OF-calls as possible.
Therefore we use the OF
as a black box and define the search
region as a (3 N - 6)-dimensional box composed
with Cartesian coordinates (x, y, z) of N atoms/molecules.
The intrinsic dimension (the number of independent variables) of
the optimization problem is reduced - when compared with 3N-dimensional
internal configurational space - due to the evident symmetry
of the objective function: the energy of the LJ-cluster depends
just on its geometry and therefore is invariant with respect to translation
(3 parameters) and rotation (3 parameters) of the cluster as a rigid
body ( see related Hessians
). Because of permutation symmetry (the atoms are assumed
identical spherical) the OF has about (N-2) ! global
extrema. Besides the nonconvex OF is known to
have O (eN'**2)
local extrema.
Thus "our" OF
is the black box which returns the OF-value provided
all the values of ( 3 N - 6 ) "controlling" parameters
from intervals [ -2.0 : 2.0 ] are given (except three
parameters: the intervals for those are chosen [ 0.0 : 2.0
] , in our case it is valid for x2,
x3 , y3
) . The analytical
form of the
OF is utilized by us for analyzing the results
exclusively.
Case: N= 8
(18-dimensional optimization problem)
A few metastable states
a)
b)
Fig. 9. The 1st example of a cluster configuration which corresponds to one of local minima: the energy of the cluster is E = -18.8286715. Some of two-bodies forces are shown with lines: red and blue lines correspond to tensional ( r < 1 ) and attractional ( r >1 ) forces respectively; a) and b) are the images of the same structure from two standpoints.
Table A9. Cartesian coordinates of the cluster.
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Table B9. Balnce of acting forces &
Binding energies
for the cluster atoms/molecules.
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a)
b)
Fig. 10. Another example of a cluster (equilibrium) configuration which corresponds to the local minima: E = -18. 8568262 . Just r < 1 are shown on b.
Table A10. Cartesian coordinates of the cluster.
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Table B10. Balance & Energies.
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The stable configuration
a)
b)
Fig. 11. The deepest equilibrium cluster
structure found with the RGA-algorithm: E = -19. 8214892.
It took in total 23
162 calls of OF
to find this cluster.
Note that "b" looks similar to that of
Fig. 9 b, but is more compact (more red
links).
Table A11. Cartesian coordinates of the cluster.
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Table B11. Balance & Energies.
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More about stability of the structures:Hessians in configurational space.
2D section of the OF (conformation energy)
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.to be continued
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