International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 3.3, pp. 379-380   | 1 | 2 |

## Section 3.3.2.2.3. Approaches to the problem of multiple minima

R. Diamonda*

aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England
Correspondence e-mail: rd10@cam.ac.uk

#### 3.3.2.2.3. Approaches to the problem of multiple minima

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Modelling methods which operate by minimizing an objective function of the coordinates (whether conformational or positional) suffer from the fact that any realistic objective function representing the potential energy of the structure is likely to have many minima in the space of the variables for any but the simplest problems. No general system has yet been devised that can ensure that the global minimum is always found in such cases, but we indicate here two approaches to this problem.

The first approach uses dynamics to escape from potential-energy minima. Molecular-mechanics simulations allow each atom to possess momentum as well as position and integrate the equations of motion, conserving the total energy. By progressively removing energy from the simulation by scaling down the momentum vectors some potential-energy minimum may be found. Conversely, a minimization of potential energy which has led to a minimum thought not to be the global minimum may be continued by introducing atomic momenta sufficient to overcome potential-energy barriers between minima, and subsequently attenuate the momenta again. In this way a number of minima may be found (Levitt & Warshel, 1975). It is equivalent to initializing a potential-energy minimization from a number of different conformations but it has the property that the minima so found are separated by energy barriers for which an upper limit is known so that the possibility exists of exploring transition pathways.

A second approach (Purisima & Scheraga, 1986) is relatively new. If the objective function to be minimized can be expressed in terms of interatomic distances, and if each atom is given coordinates in a space of dimensions for n atoms, then a starting structure may be postulated for which the interatomic distances all take their ideal values and the objective function is then necessarily at an absolute minimum. This multidimensional structure is then projected into a space of fewer dimensions, within which it is again optimized with respect to the objective function. The dimensionality of the model is thus progressively reduced until a three-dimensional model is attained at a low energy. This means that the minimum so attained in three dimensions is approached from beneath, having previously possessed a lower value in a higher-dimensional space. This, in itself, does not guarantee that the three-dimensional minimum-energy structure so found is at the global minimum, but it is not affected by energy barriers between minima in the same way, and it does appear to reach very low minima, and frequently the global one. Because it is formulated entirely in terms of interatomic distances it offers great promise for modelling molecules on the basis of data from nuclear magnetic resonance.

### References

Levitt, M. & Warshel, A. (1975). Computer simulation of protein folding. Nature (London), 253, 694–698.Google Scholar
Purisima, E. O. & Scheraga, H. A. (1986). An approach to the multiple-minima problem by relaxing dimensionality. Proc. Natl Acad. Sci. USA, 83, 2782–2786.Google Scholar