An Efficient Algorithm for Searching Low-energy Conformers of Cyclic

J. CHEM. SOC. PERKIN TRANS. 2

187

1993

An Efficient Algorithm for Searching Low-energy Conformers of Cyclic and Acycl ic Mol ecu les Hitoshi GotCa and Eiji dsawa**b Department of Chemistry, Faculty of Science, Hokkaido University, Sapporo 060, Japan Department of Knowledge- Based Information Engineering, Toyohashi University of Technology, Tempaku- cho, Toyohashi 44 I , Japan

a

A set of strategies for exhaustively finding low-energy conformers of cyclic, acyclic or alicyclic molecules is presented. Starting from any conformation, local perturbation is systematically applied t o all the flexible portions of the molecule i n question t o produce candidates o f n e w conformation. The perturbations consist of flapping and/or flipping for endocyclic bonds and stepwise rotation for acyclic bonds. The conformations they produced are believed t o lie close to the initial geometry in the conformational space. The global energy minimum (GEM) structure of the starting domain of conformational space can be quickly reached by always choosing the most stable of the conformers produced in the last perturbation cycle as the next initial structure. Once GEM o f the domain is reached, the local perturbations direct the search gradually t o higher and higher energy regions while exhaustively finding all the low-energy conformers therein. The variable search-limit strategy allows one t o use unstable conformers as the initial structure for perturbation t o ensure the exhaustiveness of the search in the low-energy region. By further increasing the search-limit, n e w domains of conformational space may be found. A program CON FLEX3 containing several additional strategies for improved performance has been tested for n-alkanes u p t o decane and cycloalkanes u p t o cyclododecane. Among the fundamental problems in molecular modelling, 'the most difficult to overcome is the global minimum problem. Even if one could exhaustively search conformational space, one still needs to correctly evaluate and rank the relative free energies of all the conformations. This is currently impossible even for systems of 100 atoms.' There is indeed a great demand for predicting the conformer distributions of large and flexible molecules, especially those of biologically interesting systems and of polymeric materials, within reasonable computer time. Whereas the prospect is bright as to the development of ultrafast computers,' really powerful software is lacking. Recent intensive research activity on the algorithms of conformational space search 3-5 and their applications reflects the situation. In practice, conformational space search is more difficult in acyclic than in cyclic molecules, the open-chain structures being generally much more flexible than the rings.* For this reason, the algorithm research in this area has primarily been directed towards cyclic s t r u c t u r e ~ , ~whereas *~ past attempts to cover acyclic conformations are sporadic at best and much less successful than the cyclic conformation ~ e a r c h . ~ This paper is a full account of our systematic study aimed at efficient coverage of low-energy regions of conformational energy space, which works equally well for cyclic and acyclic structures.' O

'

637

Algorithm Developments The known algorithms for conformational space search may be categorized into either systematic (deterministic) or random (stochastic) depending on the basic search principle, and each of them can be further divided into exhaustive or limited with regard to the extent of coverage. The obvious choice, namely the exhaustively systematic m e t h ~ d , ~ " .is' not useful, except for very small systems, because the required computer time for this method increases exponentially as the target molecule becomes larger and as the search grid is made finer. Fine grids assure a thorough search but cause

random

limited inefficiency owing to the fact that many grid points merge into the same energy minimum (redundancy problem). In order to reduce computing time, attempts have been made to remove trial structures having too short van der Waals contacts.12 However, too early pruning can be dangerous, because stable conformers sometimes emerge from unstable and seemingly insignificant conformations (uide ir~fra). Random strategies 4c~e.4'*9*13 tend to avoid the redundancy problem, at least in the early stages of the search, by performing wide-range sampling of target points from the whole conformational space. Nevertheless, as Still state^,^' the later stages of the random search suffer from the rapidly decreasing probability of finding new conformers and also from difficulties in deciding when the search is to be terminated. The most serious problem inherent in the random search is the dense population in the high energy region. Even for small acyclic systems it has recently been noticed that the high-energy space is more crowded than previously thought." Covering the conformational space of a flexible molecule is clearly a difficult task. From the above considerations, it must be concluded that a realistic approach can be neither totally systematic nor completely random. We have no other choice but to cover a limited space. An obvious choice is to search only the chemically meaningful, low-energy regions of conformational space, quickly and exhaustively. This strategy may be categorized as efficient. Our objectives are ( 1 ) to avoid going into high-energy regions, (2) to search the low-energy region exhaustively, and (3) to complete the conformation search in reasonable computer time.

188

J. CHEM. SOC. PERKIN TRANS. 2

m

1993

process

algorithm

1 2

down stream, reservoir filling, variable search-limit corner flap, edge flip, stepwise rotation, steppingstone pre-check conformational distance

Search

Inner Loop Outer Loop

3 Selection of Initial Structure

Local Perturbation

Process (1) (2)

(3) (4)

Geometry optimization

4 Comparison

Features of these algorithms are explained below. Throughout this paper, we tentatively classify the bond rotation by its dihedral angle 4 (deg) as follows:

I/O Structural data in Out lnputlsaved Initial Initial Trial Trial Optimized Optimized Stored/(none)

4 = 0 4 > 120 -120 > 4 > 0

Fig. 1 Illustration of the four elementary steps in the general conformational space search algorithm

Table I Thirty-seven rotational isomeric states of C,-molecule 1 as expressed by rotation indices at a, p and y bonds No gauche bond AAA One gauche bond GAA

G’AA

AGA

AG‘A

AGG AGG’ AGG’ G’AG

AG’G’ AG’G I AG’G G’AG’

GG’G GG‘G

GG‘G‘ GG’G‘

AAG

AAG’

G‘GG‘ G‘GG’

G’G’G G‘G‘G

Two gauche bonds GGA

I GG‘A

I GG’A GAG

G’G’A G‘A G’GA GAG‘

I

Three gauche bonds GGG GGG’ GGG‘

G‘G‘G‘ G’GG G’GG

120

Generaf Scheme.-An overview of the search algorithm is presented in Fig. 1. The four-process scheme of conformation search is actually common to most of the known conformational space search method^.^.^ It may be noted that structures are designated differently in these processes. Whole scheme contains the following nested loops: the outer loop for selecting an initial structure (process 1) and the inner loop for generating a new conformer therefrom. The latter consists of the following three steps: perturbation of initial structure to produce a candidate for a new conformation (process 2), geometry optimization of the trial structure (process 3), and comparison of the optimized structure with those already found in order to remove duplicates (process 4). For every process, new and unique algorithms have been implemented, except for the last one in the table below which is taken from Saunders: 4d