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RGA - algorithm for black box global optimization

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.
 X Y Z 1 -0.79212E+00 0.38599E+00 0.37373E+00 2 0.74053E+00 0.00000E+00 0.00000E+00 3 0.63796E-01 0.73243E+00 0.00000E+00 4 -0.67481E+00 -0.57931E+00 0.61308E+00 5 -0.22312E+00 -0.20538E+00 -0.18497E+00 6 0.78930E-01 0.57610E-01 0.73405E+00 7 0.49073E+00 -0.68082E+00 -0.68007E+00 8 0.31606E+00 0.28948E+00 -0.85582E+00

Table  B9.  Balnce of acting forces  & Binding energies
for the cluster atoms/molecules.
 1 3.97342e-06 -4.23076 2 4.84257e-06 -5.19763 3 8.24184e-06 -5.20911 4 1.53265e-06 -3.30113 5 5.75157e-07 -6.98914 6 8.9605e-06 -5.19766 7 4.79571e-07 -3.30115 8 8.39392e-06 -4.23076

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.
 X Y Z 1 -0.47329E+00 -0.46362E+00 0.83369E+00 2 0.87991E+00 0.00000E+00 0.00000E+00 3 -0.47354E+00 0.95384E+00 0.00000E+00 4 0.73348E-01 0.29247E+00 0.49721E+00 5 0.60052E+00 -0.48633E+00 -0.82711E+00 6 - 0.74316E+00 -0.49481E-02 -0.82610E-02 7 0.68073E-01 0.28668E+00 -0.50204E+00 8 0.68137E-01 -0.57809E+00 0.65074E-02

Table  B10.   Balance & Energies.
 1 1.5463e-07 -3.32797 2 3.2774e-07 -4.29217 3 6.10375e-07 -3.32798 4 5.67575e-07 -6.09744 5 1.40072e-07 -3.26687 6 1.41823e-07 -5.19766 7 3.23987e-07 -3.30115 8 5.09239e-07 -4.23076

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.
 X Y Z 1 -0.14563E+00 -0.12550E+00 0.37908E+00 2 0.76767E+00 0.00000E+00 0.00000E+00 3 0.11327E+00 0.75927E+00 0.00000E+00 4 0.51290E+00 0.44205E+00 0.85575E+00 5 -0.31102E-02 -0.26808E-02 -0.62338E+00 6 -0.70804E+00 -0.61026E+00 -0.27821E+00 7 0.26047E+00 -0.84521E+00 -0.16662E+00 8 -0.79753E+00 0.38233E+00 -0.16663E+00

Table  B11.   Balance & Energies.

 1 3.01559e-06 -6.97399 2 7.01741e-07 -5.2109 3 5.00438e-07 -5.2109 4 1.36032e-07 -3.31897 5 2.36973e-07 -6.08125 6 6.17512e-07 -4.24293 7 6.65146e-08 -4.30201 8 4.18526e-07 -4.30202

More about stability of the structures:Hessians  in configurational space.

2D section of the OF (conformation energy)

Fig. 12.  The energy landscape:   2-D section of the objective function ,  which involves all of 3 structures displayed above.  V1 and V2  represent distances from the global minimum point in the 18-dimensional box  along  two basic vectors  V1 and V2 respectively. Points of discontinuity of the OF are not shown: the  OF-values, which are greater than 8.5,  are cut off.

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