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Tabu Search Based Strategies for Conformational Search†
Svetlana Stepanenko and Bernd Engels*
JuliusMaximilliansUniVersita ¨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 descentmodest 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.
1. Introduction
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 forcefield parameters
or the determination of possible reaction paths between reactants
and products.24The 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
room temperature.712
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,1723each 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
literature.3436
In the present paper, we utilize a variant of the tabu search
(TS) for conformational search. The TS3739is a metaheuristic4043
which employs the “steepest descentmodest 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
plied.4449
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.5061We 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 quasiNewton methods is used.6467
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: Email: bernd@
chemie.uniwuerzburg.de. Phone: (+49)9318885394. Fax: (+49)931888
4606.
J. Phys. Chem. A 2009, 113, 11699–11705
11699
10.1021/jp9028084 CCC: $40.75
2009 American Chemical Society
Published on Web 09/21/2009
Page 2
the GTS, but to escape a local minimum via the modest ascent,
only a grid of function evaluations is employed. The TSPA uses
the same method for the modest ascent strategy as the GOTS,
but Powell’s algorithm is employed for the steepest descent part.
Due to the different strategies, the TSPA only needs functional
evaluations but no gradients or Hessians, while the GOTS only
requires gradients. All other concepts used within the GOTS
and the TSPA were taken from the GTS. Numerical results for
the GTS and GOTS methods, which are available in ref 63,
showed that these approaches are more efficient than previous
global optimizers. For functions with higher dimensionality, the
GOTS becomes more efficient than the GTS. The efficiency of
the TSPA method is comparable to that of genetic algorithms.69,70
In the present paper, the GOTS has been adapted for
conformational searches. To increase the efficiency of the
optimization, our first ansatz has been combined with simulated
annealing elements.7174The method was implemented into the
computational chemistry environment ChemShell, and test
computations have been performed for lysine and arginine, two
ACE (angiotensin converting enzyme) inhibitors, 2acetoxy
N,N,Ntrimethylethanaminium (acetylcholine), and an HIV1
protease inhibitor. The paper is organized as follows. In section
2, various details of the used algorithms are described. Then,
suitability of the new approach is tested using some examples.
It is described in section 3. Conclusions complete the work.
2. Adaptation of the GOTS Algorithm for
Conformational Search
The choice of the coordinates used to describe a molecule is
critical for the efficiency of the optimization.63Since gradients
and Hessians are usually calculated in Cartesian space, Cartesian
coordinates would be the most straightforward choice. However,
transitions from one conformer to another one are mainly
accompanied by variations in dihedral angles, while bond
distances and angles change only slightly. To exploit this
difference in the stiffness, we employed nonredundant internal
coordinates and varied only the dihedral angles to escape a local
minimum. During minimization to the next local minimum, all
coordinates are optimized. The nonredundant coordinates were
represented in the Zmatrix formalism employed in the Gaussian
program package.75
Often, various dihedral angles have to be changed together
to perform proper rotations around a given single bond. For
example, to rotate a terminal methyl group of the nbutanol
molecule (see Figure 1) consisting of atoms C1H6H7H8around
the C1C2axis, it is necessary to modify three dihedral angles
(C3C2C1H6, C3C2C1H7, C3C2C1H8) at the same time.
According to Chass76or Echenique and Alonso,77such proper
rotations can be assured by dividing the dihedral angles into
main and dependent torsions.78,79Hence, during the modest
ascent part, only the main torsions are independently varied,
while the dependent torsions are modified accordingly so that
proper rotations take place. To find the next local minimum,
all internal coordinates (distances, angles, and dihedral angles)
are again independently optimized. The algorithm used to
determine main and dependent dihedral angles is depicted in
Figure 2.
During the modest ascent search, the GOTS algorithm uses
only function evaluations to escape a local minimum. At the
local minimum, all functional values Fzi+and Fzi(eq 2) are
computed with singlepoint calculations to determine the general
direction of the following moves. They are obtained by varying
each dimension by a user defined step size ∆xi0. In our previous
approach, the direction in which the search is moved is estimated
according to62
Here, xidenotes the ith coordinate. In our previous approach,
the coefficients rankmaxand rankminwere set to 0 and 1. However,
for the conformational search, the computed energy differences
were found to be too small to obtain reasonable ranking
coefficients ranki. Hence, the coefficients rankmaxand rankmin
are calculated for each case separately:
Figure 1. Atom numeration of the nbutanol molecule.
Figure 2. Flowchart of the determination of main and dependent
torsion angles.
xi
new) xi
old+ ∆xi
0× ranki
(1)
Fzi
Fzi
+) F(x1,x2,... ,xj+ ∆xj
) F(x1,x2,... ,xj ∆xj
0,... ,xNDIM)
0,... ,xNDIM)
(2)
Di){
+, if Fzi
, if Fzi
+< Fzi
+> Fzi


(3)
Fzi) F(x1,x2,... ,xj+ Dj× ∆xj
0,... ,xNDIM)
(4)
ranki) rankmin+ (rankmax rankmin)(
Fzmin Fzi
Fzmax Fzmin)
(5)
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J. Phys. Chem. A, Vol. 113, No. 43, 2009
Stepanenko and Engels
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To calculate rankmax, the same percentage analysis as for rankmin
is made for each value:
The resulting modest ascent is followed until the next
calculated function value is smaller than the previous one. This
indicates that the barrier to the next valley is crossed. From
this point, the next local minimum is located using the local
minimization routines.
Reverse moves and cycles were avoided by the tabu list (TL)
concept as used previously. For the conformational search,
however, only tabu regions (TR) could be employed, since tabu
direction (TD) turned out to be incompatible. A diversification
search (DS) is performed if the search gets stuck in an
unpromising area. For the conformational search, this is
performed if the solution does not improve after a given number
of new local minima (parameter BADMAX, see Table 1) or if
all neighborhood solutions of the local minimum under con
sideration are already set tabu. To ensure that the search switches
to regions that were not already investigated in the DS, the main
torsions are changed in steps of 60° (parameter diversification
step, see Table 1) and the dependent torsions are moved
accordingly. The rest of the variables are kept constant. After
excluding all points which belong to an already established TR,
the new search starts from the point with the lowest energy
value.
To include more probability aspects, ideas from the simulated
annealing (SA) method have been implemented into the GOTS.
They shall help to focus on promising areas of the large search
space. In our approach, the SA ideas are exploited to determine
if a new minimum is taken as the new starting point or not. If
the new minimum is lower in energy, it is always taken. If it is
higher than the previous one, the Metropolis criteria80is used
to decide if it is nevertheless taken as the new starting point. If
the new minimum is not accepted, the algorithm returns to the
previous minimum and continues the search along the next
modest ascent direction. This SA element directs the search to
more promising areas. If the next local minimum is very high,
the search concentrates on the region around the last minimum
which is considerably lower in energy. The flowchart given in
Figure 3 depicts the GOTS approach to the conformational
search.
The GOTS possesses some userdefined parameters which
are summarized in Table 1. The parameter ∆xigives the step
size during the modest ascent search. One could assume that
small step sizes increase the accuracy. However, this is
misleading, since the energy differences between the directions
get so small that the ranking procedure does not work properly
anymore. On the basis of our experiences, ∆xi ) 45° is
recommended as a standard value. With larger step sizes, the
average number of steps needed to leave a local minimum
decreases but sometimes minima are missed.
In our previous GOTS version, a diversification search is
started if the solution does not improve after BADMAX local
minima. As in previous versions of the GOTS, BADMAX was
set to 5, but in the present version, this hard rule is modified by
the Metropolis criterion for which the parameter T is used. In
these first test cases, T is set 150 kJ mol1. With such high T
values, the probability that high lying minima are taken as the
new starting structure is also high. Hence, this setting ensures
that also high lying minima are used as new starting points.
This is necessary to overcome high lying regions and to move
to complete different regions. With lower T values, the GOTS
could get trapped in the given region. The value of T is
decreased by 10% for each newly found minimum which is
higher in energy.
The parameter TR controls the size of the tabu regions within
the whole optimization process. One would expect that small
TR do not efficiently block already visited regions so that the
effort decreases with increasing TR. However, in most cases,
the effort increases for larger TR. This counterintuitive behavior
may result because the tabu regions become so large that minima
lying close by already visited points are overlooked or because
the optimal path to the global minimum is blocked. A value of
TR ) 10°, however, seems to be a good choice for a large
variety of different problems.
3. Applications
To study its suitability for conformation searches, the
modified GOTS was applied to conformational studies for lysine,
rankmin) 2 
rankmin) 0.1
Fzmax
Fzmin
if Fzmax< 2 · Fzmin
otherwise
(6)
rankmax) 2 
rankmax) 1.0
Fzi
Fzmin
if Fzi< 2 · Fzmin
otherwise
(7)
TABLE 1: Description of the Parameters of the GOTS
Application to the Conformational Search
parameterpurpose
recommended
values
∆xi
step size at the modest ascent
strategy
number of not improved
minima after which a DS is
performed
45°
BADMAX
5
diversification step step size used during the
diversification strategy
rankmax
default maximum recency
ranked value
rankmin
default minimum recency
ranked value
TR
T
control parameter (analog of
temperature)
60°
1.0
0
1/2of the tabu region diameter10°
150 kJ mol1
Figure 3. Flowchart of GOTS with SA elements.
GOTS Based Strategies for Conformational Search
J. Phys. Chem. A, Vol. 113, No. 43, 2009 11701
Page 4
arginine, 2acetoxyN,N,Ntrimethylethanaminium (acetylcho
line), the ACE inhibitor 1, Fosinopril which represents another
ACE inhibitor, and the HIV1 protease inhibitor (2). All
computations were performed within the ChemShell environ
ment running under Linux. For the calculations, the universal
force field (UFF)81was employed. Lysine has 66 internal degrees
of freedom which can be built up with 21 dihedral angles.
According to our approach, these dihedral angles are subdivided
into 8 main torsions and 13 dependent torsions. Hence, during
the modest ascent part (escape from a local minimum) for lysine
only 8 torsions are varied while all internal degrees of freedom
are optimized within the local minimization part (finding the
next local minimum). The corresponding numbers of main
torsions are 11 for arginine, 8 for acetylcholine, 17 for the ACE
inhibitors 1, 34 for Fosinopril, and 41 for the HIV1 protease
inhibitor 2. They are indicated in Figures 4 and 5.
Let us first concentrate on the smaller molecules lysine,
arginine, and acetylcholine. The main torsions of lysine are
indicated in Figure 4a (upper panel). To test how divideand
conquer (DaC) strategies82accelerate the convergence of the
GOTS, the main torsions were further subdivided into those
connected with the residue (Figure 4, upper panel b) and those
connected with the backbone part, i.e., the carboxylic acid,
the ammonium, and the Rcarbon center (Figure 4, upper panel
c). The subdivision of the main torsions of arginine and
acetylcholine is also indicated in Figure 4 (middle and lower
parts). In the DaC run, the conformational search starts with
the smaller part. Then, the best solution is kept constant within
the modest ascent part of the consecutive search. During the
local minimization part, again all internal coordinates are
optimized without any restriction. Since within the local
minimization part all internal degrees of freedom are optimized,
the first minimum found in the normal approach (no subdivision
of main torsions) is the same as that found within the DaC
strategy.
The results for the smaller molecules are summarized in Table
2. The runs were started independently from different structures
to test the stability of the GOTS. Table 2 gives the energies of
the starting structures, of the first minimum encountered in the
GOTS, and of the lowest lying structures found in one GOTS
run. All energies are given relative to the lowest minimum found
in the all GOTS performed for the molecule. The structures
found in the GOTS which started from start structures 1 are
illustrated in Figures 68. Finally, Figure 9 monitors the
progression of conformational search starting from start struc
tures 1.
Results show that despite similar sizes lysine and acetylcho
line seem to represent easier examples than arginine. For the
former, the lowest lying minimum is found independently of
the starting structure, although for lysine the DaC strategy seems
to be favorable in comparison to the normal approach (no
subdivision of main torsions). Figures 9 and 10 show addition
ally that the GOTS in the present form indeed scans large parts
of the phase space and is able to locate low lying as well as
high lying minima. For lysine, the energies of the detected
minima vary over a range of 100 kJ mol1. For acetylcholine,
the energies of the detected minima alter in a range of 25 kJ
mol1. Please note that the tabu list (TL) impedes that the same
minimum appears twice in Figure 9; i.e., despite very similar
energies, all minima correspond to different conformers (e.g.,
minima 2326 for lysine or minima 10, 12, 14, 15, etc., for
acetylcholine). This indicates that the GOTS is also able to find
energetically close lying minima which is, for example, of
interest for the determination of Boltzmannweighted ensembles.
All results show that arginine represents a more difficult
example than lysine or acetylcholine. In all cases, very low lying
Figure 4. Main torsions of lysine (upper part), arginine (middle part),
and acetylcholine (lower part) and their subdivision used in the divide
andconquer (DaC) strategy.
Figure 5. Main torsions of the ACE inhibitor 1, Fosinopril, and the
HIV1 protease inhibitor (2).
TABLE 2: Test Results for Lysine, Arginine, and
Acetylcholinea
lowestc
molecule start structurefirstb
normalDaC
Lys1d
2
3
4
1d
2
3
4
1d
2
3
1323e
486
306
399
1452
1398
1385
1892
309
716
4735
13
13
0.3
0.8
0.0
0.0
2.9
6.3
5.5
9.2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.5
0.0
1.8
0.0
0.0
0.0
7
8
Arg
58
49
44
49
24
19
19
acetylcholine
aAll energies are given with respect to the energetically lowest
minimum (kJ mol1).bFirst minimum found in the conformational
search.cLowest lying minimum found in the conformational search.
dStructures obtained from starting structure 1 are illustrated in
Figures 68.eEnergies relative to the lowest minimum.
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Stepanenko and Engels
Page 5
minima are detected; however, the apparent global minimum
is only found if the DaC strategy is used. Figure 9 also shows
that during the GOTS run high lying minima are encountered
more often than for lysine or acetylcholine. Nevertheless, the
energies of the detected minima vary in the same range as for
lysine (70 kJ mol1).
To test the suitability of our approach for larger molecules,
conformational searches were also carried out for Fosinopril,
the ACE inhibitor 1, and the HIV1 protease inhibitor 2 (Figure
5). DaC strategies were also tested for these molecules, but with
or without such strategies, the same global minimum was
detected and also the convergence of the conformational search
was about the same. Since the DaC tests did not differ
considerably from the normal approach, Table 3 only gives the
results of the normal approach. Figure 11 gives the geometrical
arrangements of the starting structure and of the first and lowest
lying conformer found for 1, and Figure 10 shows the progres
sion of the corresponding conformational search.
The results obtained for the larger molecules show that also
for larger molecules the GOTS is able to locate various local
minima with quite different nuclear arrangements and quite
varying energies. As found in previous applications, the GOTS
very fast leads to solutions close to the global minimum but is
also able to scan large parts of the phase space. Important for
Figure 6. Illustration of the structures characterized in Table 2 for
lysine.
Figure 7. Illustration of the structures characterized in Table 2 for
arginine.
Figure 8. Illustration of the structures characterized in Table 2 for
acetylcholine.
TABLE 3: Test Results for the Larger Moleculesa
molecule start structurefirst minimum lowest minimum
Fosinopril
1
2
676
758
665
47
29
74
0
0
0
aAll energies are given with respect to the energetically lowest
minimum (kJ mol1).
Figure 9. Progression of conformational search starting from start
structure 1.
GOTS Based Strategies for Conformational Search
J. Phys. Chem. A, Vol. 113, No. 43, 2009 11703
Page 6
this is the ability of the GOTS to surmount high lying regions
as for example shown in Figure 10 for compound 1 (minima
1222).
4. Conclusions and Future Work
In this paper, the global optimization routine called gradient
only tabu search (GOTS) was adapted and applied to confor
mational search problems. The paper describes various algo
rithmic details, such as refinement of ranking procedure of the
original GOTS method, the definition of the variables, and the
exploitation of simulated annealing elements. Parameters of
the approach are discussed, and appropriate values are recom
mended. To test the suitability of the approach for conforma
tional search, various molecules of differing size are used as
test cases. The results indicate that the GOTS is able to detect
the low lying minima of molecules with a high number of freely
rotatable single bonds, since its strategy directs the search to
low lying parts of the potential energy surface. Nevertheless,
since higher lying minima are also accepted, the search is also
able to surmount high lying parts. This ability makes the GOTS
less dependent on the starting structure, since it enables the
traverse of high lying regions which separate two low lying
ones with quite differing nuclei arrangements. Future work will
concentrate on algorithms which provide promising starting
structures, e.g., conformations which are stabilized by internal
hydrogen bonds.
Acknowledgment. Ongoing financial support by the Deutsche
Forschungsgesellschaft in the framework of the SPP 1178 and
the SFB 630 is gratefully acknowledged.
Supporting Information Available: Figures showing struc
tures of Fosinopril and HIV1 protease inhibitor. This material
is available free of charge via the Internet at http://pubs.acs.org.
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