The Multiple Minima Problem

In many applications of molecular mechanics it is important that the lowest energy structure, i.e., the global energy minimum, be found. This is particularly true when one would like to model the structures that are predominant in solution, or establish the distribution of all the conformers1161. 

An excellent comparison of the methods employed in conformational searching of organic molecules has been published11061. In the vast majority of published inorganic molecular modeling studies no stochastic or molecular dynamics conformational searches have been conducted. There are three reasons for this. 

There have, however, been a number of recent studies involving conformational searching of metal complexes. The geometries of flexible side-chains have been 

A number of studies have been conducted, however, involving the conformational searching of metal complexes. The geometries of flexible side chains have been investigated using molecular dynamics [247], and conformational searching using Monte Carlo and molecular dynamics methods has been carried out on transition metal complexes [72] and metalloproteins [248]. 

A related problem, and one that is commonly encountered, has to do with molecules possessing multiple conformations. Consider A-rnclhylacclamide, which can exist in E and Z forms. The latter stereoisomer is favored over the former by about 3 kcal/mol, but the barrier 

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Piela L, Kostrowicki J and Scheraga H A 1989 The multiple-minima problem in the conformational analysis of molecules. Deformation of the potential energy hypersurface by the diffusion equation method J. Phys. Chem. 93 3339... 

Q and H A Scheraga 1987. Monte-Carlo-minimization Approach to the Multiple-minima Problem... 

Li Z Q and H A Scheraga 1987. Monte Carlo Minimization Approach to the Multiple Minima Problem, in Protein Folding. Proceedings of the National Academy of Sciences USA 84 6611-6615. 

Purisima, E.O. Scheraga, H.A., An approach to the multiple-minima problem by relaxing dimensionality, Proc. Natl Acad. Sci. USA 1986, 83, 2782-2786... 

Cvijovic, D. and Klinowski, J. (1995) Taboo search—an approach to the multiple minima problem. Science 267, 664—666. 

Z. Li and H. A. Scheraga. Monte carlo minimization approach to the multiple minima problem in protein folding. Proc. Nat. Acad. Sci. U.S.A., 84 6611-6615, 1987. 

Challenging the Multiple Minima Problem Example of Protein Folding... 

The multiple minima problem is ubiquitous in mathematical description of reality (physics, chemistry, biology, economy, etc.), and (R) is only an example of it. It is very unlikely that such kind of problems will find analytical solution, and thus, one has to rely on computers. The hope is that for certain class of approximations one can simplify (R) enough to overcome the multiple minima problem by extensive calculations. 

The protein folding, notorious for an astronomic number of possible conformations, is only an example of the multiple minima problem, inherently connected to all applications of theory to structural chemistry (isomers, supramolecular structures etc.). The multiple minima problem is also virtually ubiquitous in other sciences, and whenever a mathematical description is used, the situation is encountered more and more often. Despite the complexity of the protein folding, remarkable achievements in the prediction of the 3D structure of globular proteins are possible nowadays. 

It is instructive to see what kind of concepts have been used to overcome the multiple minima problem in this particular case, with a hope that a similar approach could help in other domains. Here is the list of decisive steps ... 

L. Piela, H.A. Scheraga, On the multiple-minima problem in the conformational analysis of polypeptides. I. Backbone degrees of freedom for a perturbed a-helix. Biopolymers 26, S33-S58 (1987)... 

In this article, we have reviewed some of powerful generalized-ensemble algorithms for both Monte Carlo simulations and molecular dynamics simulations. A simulation in generalized ensemble realizes a random walk in potential energy space, alleviating the multiple-minima problem that is a common difficulty in simulations of complex systems with many degrees of freedom. 

The task of minimizing potential energy functions arising in molecular mechanics is typical of optimization applications seeking favorable configurational states of a physical system. 18-23 The sheer size of configuration space and complexity of the system introduce two major problems extensive computational requirements and the multiple-minima problem. 

L. Piela, J. Kostrowicki, and H. A. Scheraga,/. Phys. Chem., 93,3339 (1989). The Multiple-Minima Problem in Conformational Analysis of Molecules. Deformation of the Potential Energy Hypersurface by the Diffusion Equation Method. 

E. O. Purisima and H. A. Scheraga, Proc. Natl. Acad. Sci. USA, 83, 2782 (1986). An Approach to the Multiple-Minima Problem by Relaxing Dimensionality. 

Finally, with all the current activity on the multiple-minima problem, which is essentially solved for oligopeptides and regular-repeating structures of fibrous proteins, we may hope to see further progress in treating larger molecules, that is, globular proteins, by more efficient searches of conformational space. 

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