CONCLUDING REMARKS STOCHASTIC MOLECULAR OPTIMIZATION USING THOR - MOLECULAR FORCE

 

APPLICATIONS

We have applied the SMO to investigate conformations of some molecular systems. The first application of our approach is to search the range of that answer the question ''Which or distribution of probability correspond to a minimum number of steps to find the global minimum energy ?''.

To answer this question we have generate two maps (Figs. 2-3) in terms of nC is the number of cycles to obtain the convergence in the energy. That is, for each (q,T) we calculate the number of steps necessary to search the global minimum in case of H O and H O molecules.

H₂O (Dimension D=1)

Using fixed H-O distance we scan the angle (D=1) between H-O-H bonds to investigate the angular energy (rotational barrier) for the H₂O molecules. If we scan the potential surface in 0.1 steps this results in 3600 cycles necessary to search the angular range (0 360 ). However, with the random procedure proposed by GSA algorithm we obtain the global minimum with less 180 cycles for (q ; q )=(1.5; 1.7) machine (Fig. 2). When q is close to 1.7 the algorithm convergence is faster than other q values.

H₂O₃ (Dimension D=2)

For the H O molecule we have optimized two (D=2) proper dihedral angles and (Fig. 3 - 4) and obtained the global minimum with less then 840 cycles using (q ; q )=(1.5; 1.6) machine (we point out that 3600x3600 cycles is far greater than 840 cycles).

The figure 3 shows the range of the q parameter. When q is close to 1.6 the algorithm convergence is faster than other q values. This behavior is obtained for the H₂O₃ molecule when we optimize H-O-O angle.

Both results, in case of equilibrium geometry, according qualitatively which a semi-empirical procedure [5], however we have obtained all mapping using SMO faster than semi-empirical one.

Other important result is that the variation range of q (Fig. 2-3) agree with the relationship proposed in reference [14], that is;

 

That equation defines the region where q is optimized, wich corresponds to a minimum number of steps to find the global minimum. D is the space dimension or the number of parameters to be optimized.

POLYPEPTIDES (Dimension D=36)

As a second application we have used the SMO approach to verify the formation of -helix structure in polypeptides. Other goal of this application is to check the potentiality of the SMO method in case of large molecular system or in case o large dimension D. We investigate two structures: pentalanin and icoalanin molecules. These are polypeptides molecules, composed by five and twenty alanin residues, respectively.

The search of the global minimum by the SMO approach allow us to map the hypersurface's conformational energy, as shown in Figures 4 and 5 for the H O and icoalanin molecules, respectively.

The helix (a polypeptide structure) is a rodlike structure. The tightly coiled polypeptide main chain forms the inner part of the rod, and the side chain extends outward in a helical array in low dielectric medium. The helix is stabilized by hydrogen bonds between the NH and CO groups of the main chain. The CO group of each amino acid is hydrogen bonded to the NH group of the amino acid that is situated four residues ahead in the linear sequence. Thus, all the main-chain CO and NH groups are hydrogen bonded. We have studied these compounds, optimizing the dihedral angles: (CH C-N-CH ), (N-CH C-N) and (C- N-CH C).

We show that pentalanin does not form stable helix structure, probably because the number of hydrogen bonds are insufficient to maintain the helical array. We optimized the , and dihedral angles and obtained, using SMO approach, a global minimum which does not correspond to the helical array,.also the ''most frequent'' , and dihedral angles found in the THOR-GSA run does not correspond to helix structures.

On the contrary, the icoalanin molecule is an helix structure. The following improper angles are expected for an helix structure: = 180 , = - 40 and = - 60 . The torsion angle has two minima. One at 0 and the other one at 180 . The later value is always observed in biological structures, except for special situations for proline residues. In order to simplify the calculation we fix the = 180 . Figure 5 shows the , conformation energy surface obtained. This figure shows, as expected, that the global minimum occurs at the angles = -60 and = -40 .

We can obtain the helix structure, simulating all , pairs, in a few minutes using an IBM risc-6000 workstation. With q =1.5, q = 2.0 and T=100 the simulation takes 3673 to 100000 cycles for 18 , pairs (D=36). Meanwhile, using MD, we would have waited on a few days of simulation!! Further, depending on the initial condition, we do not obtaining an helix structure since MD only guarantees a dynamical structure which is frequently a local minimum, because protein folding is not in MD time scale.

We would like to point out, in order to compare the SMO method with the usual Monte Carlo one, that a similar simulation was done in reference [16]. In this last case the folding of helix is reached in 100 millions of cycles, which makes the SMO a promising method for protein folding studies.

All force field parameters used in the applications presented in this paper, are listed in the tables I. a.,b ,c,d,e, The used atomic nomenclature is H (hydrogen), HO (hydroxyl hydrogen), CH (aliphatic CH-group), CH (aliphatic CH -group), N (nitrogen), NT (terminal nitrogen), O (oxygen), OA (hydroxyl oxygen). \

CONCLUDING REMARKS STOCHASTIC MOLECULAR OPTIMIZATION USING THOR - MOLECULAR FORCE