Tabu Search Based Strategies for Conformational Search†
Svetlana Stepanenko and Bernd Engels*
Julius-Maximillians-UniVersita ¨t Wu ¨rzburg, Institut fu ¨r Organische Chemie, Am Hubland,
97074 Wu ¨rzburg, Germany
ReceiVed: March 28, 2009; ReVised Manuscript ReceiVed: August 12, 2009
This paper presents an application of the new nonlinear global optimization routine gradient only tabu search
(GOTS) to conformational search problems. It is based on the tabu search strategy which tries to determine
the global minimum of a function by the steepest descent-modest ascent strategy. The refinement of ranking
procedure of the original GOTS method and the exploitation of simulated annealing elements are described,
and the modifications of the GOTS algorithm necessary to adopt it to conformation searches are explained.
The utility of the GOTS for conformational search problems is tested using various examples.
Efficient searches for global minima of highly dimensional
functions1with numerous local minima are central for the
solution of many problems in computational chemistry. Well
known examples are the optimization of force-field parameters
or the determination of possible reaction paths between reactants
and products.2-4The identification of the energetically lowest
lying conformers of molecules possessing a high number of
freely rotatable bonds is another important global optimization
problem in computational chemistry.5,6These conformers are
important, since they determine most molecular properties at
Mathematically, such a conformational search represents a
global optimization problem in which the potential energy
function of the molecule is the objective function while the
coordinates, that are used to represent the conformation of the
molecule, are the variables. The perfect global optimization
routine would always give the shortest way from a given starting
point to the global minimum. For the potential energy surface
(PES) of a molecule, this way includes downhill and uphill
moves. While the best downhill moves can be well approximated
as the shortest way to the next local minimum, the uphill moves
are less straightforward, since the direction to the global
minimum is not known in advance. For smaller molecules, the
global energy minimum and the lowest lying minima can be
determined systematically.13,14One possibility is to choose a
large number of starting conformations that are equally distrib-
uted on the energy surface. From each of them, a minimization
to the nearest minimum is performed using local optimization
techniques and then all duplicated structures are rejected.15With
an increasing number of freely rotatable single bonds, however,
the search space increases strongly with the number of degrees
of freedom (e.g., torsion angle), which is typically proportional
to the size of the molecule. This is known as combinatorial
explosion.16Therefore, to obtain the energetically low lying
conformations at tractable computational cost, specialized
conformational search algorithms are needed. Over the past
several years, a multitude of conformational search techniques
have been developed for this purpose,17-23each with its
particular strengths and weaknesses. Reviews about commonly
used techniques, such as classical molecular dynamics simula-
tion (MD),24,25mutually orthogonal Latin squares (MOLS)
conformational search technique,26smoothing/deformation27and
systematic search methods,28,29Monte Carlo,30simulated
annealing,31,32and genetic algorithms,33can be taken from the
In the present paper, we utilize a variant of the tabu search
(TS) for conformational search. The TS37-39is a metaheuristic40-43
which employs the “steepest descent-modest ascent” strategy.
The steepest descent is taken to find the next local minimum,
while the modest ascent path is followed to escape a local
minimum and to search for the next local minimum. Reverse
modes and cycles are prevented by the use of a tabu list (TL)
which sets already visited solutions tabu. The TL also recognizes
if the search gets stuck in a given region. In such cases, a
diversification search (DS) is performed which guides the search
to different and hopefully more promising regions of the search
space. For many applications in a wide variety of fields, the TS
yielded much better solutions than methods previously ap-
An adaptation of the used algorithm to the computational
chemistry problems mentioned above is not straightforward,
since, for example, conformational searches or the optimizations
of force field parameters represent continuous optimization
problems. Nevertheless, several attempts have been made to deal
with continuous optimization problems.50-61We developed three
different approaches, the gradient tabu search (GTS),62the
gradient only tabu search (GOTS),63and the tabu search with
Powell’s algorithm (TSPA).63The GTS algorithm uses analytical
gradients for a fast minimization to the next local minimum
and the diagonal elements of the analytical Hessian to escape
local minima. For the minimization, a combination of the
steepest descent and the quasi-Newton methods is used.64-67
To follow the modest ascent, the diagonal elements of the
Hessian are employed. To determine the direction, they are
weighted by a linear ranking procedure. For the tabu list,
concepts as tabu directions (TD) and tabu regions (TR) which
are related to previous ideas of Glover68were used to ensure
an efficient blocking of already visited regions. GOTS and TSPA
were developed to avoid the computation of Hessian and also
of gradients, respectively. To determine the next local minimum
(local minimization part), the GOTS uses the same strategy as
†Part of the “Walter Thiel Festschrift”.
* Author to whom correspondence should be addressed: E-mail: bernd@
chemie.uni-wuerzburg.de. Phone: (+49)931-888-5394. Fax: (+49)931-888-
J. Phys. Chem. A 2009, 113, 11699–11705
10.1021/jp9028084 CCC: $40.75
2009 American Chemical Society
Published on Web 09/21/2009
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GOTS Based Strategies for Conformational Search
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