Discovering Unique, Low-energy
Transition States of Small, Non-cyclic
Molecules Using Evolutionary
Molecular Memetic Computing
M.M.H. Ellabaan1, Y.S. Ong2, S.D. Handoko2, C.K. Kwoh3, and H. Man4
1
Center for Systems Microbiology, Department of Systems Biology, Technical University of Denmark, Lyngby, Denmark
Centre for Computational Intelligence, School of Computer Engineering, Nanyang Technological University, Singapore
3
BioInformatics Research Centre, School of Computer Engineering, Nanyang Technological University, Singapore
4
Department of Biology, Boston University, Boston, MA, USA
2
Abstract—In the last few decades, identification of the transition states has experienced significant
growth in research interests from various scientific communities. As per the transition states theory,
reaction paths and landscape analysis as well as many thermodynamic properties of biochemical
systems can be accurately identified through the transition states. Transition states describe the
paths of molecular systems in transiting across stable states. In this article, we present a novel
Molecular Memetic Computing (MMC) methodology for the discovery of unique, low-energy
transition states and showcase the efficacy of identifying the transition states using the evolutionary
nature of the memetic computing paradigm. In essence, the MMC is equipped with the tree-based
representation of non-cyclic molecules and the covalent-bond-driven evolutionary operators in
addition to the typical backbone of memetic algorithms. Herein, we employ genetic algorithm for
the global search, Berny algorithm for the individual learning, and make use of the valley-adaptive
clearing scheme as the niching strategy in the spirit of Lamarckian learning. Experiments with a
number of small non-cyclic molecules demonstrated excellent efficacy of the MMC compared to
recent advances of several state-of-the-art algorithms. Not only did the MMC uncover the largest
number of transition states, but it also incurred the least amount of computational costs.
1. Introduction
Small molecules with a few dozens of atoms play important roles in biology, chemistry, and
medicine. Alongside their usage as the combinatorial building blocks in chemical synthesis, small
molecules have become essential for screening, designing, discovering, and synthesizing new drugs
[1]. Discovery of the low-energy stereoisomers of small molecules has gained growing attention of
the scientific community in the past decades as these stereoisomers may lead to the discovery of
more effective drug molecules compared to their traditional counterparts [2–4]. Such stereoisomers
are not always available in nature. They may require special synthesis, for which knowledge on their
transition states are often required in order to identify the minimum energy pathway to achieve the
synthesis. The transition states describe molecular configurations assumed during the synthesis of a
stereoisomer from another stereoisomer. Transient nature of the transition states makes
computational study a necessity as it is difficult—if not impossible—to observe them through wetlab experiments. Knowledge on the transition states helps scientists not only in the synthesis of drug
isomers, but also in the identification of drug sensitivities to environmental changes. The lower the
energy of the transition states associated with a given drug isomer, the more vulnerable will the
drug be to the environmental changes. Such information may be used to measure the relative ranks
of drug analogs. Analogs that are less sensitive to environmental changes can then be recommended
subject to pathological conditions.
Identification of the transition states is a computationally challenging task. Given a small molecule,
its energy landscape is generally highly constrained and involves vast search space. While no explicit
constraint function is normally formulated, the energy terms that serve as the objective function can
easily approach infinity, suggesting the nonsensical or the naturally impossible atomic configuration
of the molecule. Furthermore, the energy calculation of the molecule is generally computationally
expensive. With a single evaluation requiring minutes to hours of CPU time, the challenge is hence to
maximize the production of meaningful conformations so as not to waste the precious
computational resources evaluating many nonsensical structures. To-date, many optimization
approaches have been proposed to identify the transition states. These can be classified into the
conventional and the stochastic approaches [5].
Conventional approaches assume the availability of domain knowledge to produce reasonably good
initial guesses of the transition states. Methods in this category can be further segregated into
double-ended and single-ended techniques [6]. Given pairs of known stereoisomers (reactants and
products), the double-ended methods find transition states that bridges them [7–9]. In contrast, the
single-ended methods find transition states in the vicinity of some initial guesses [10]. Upon
availability of such domain-specific a priori information, conventional approaches converge
efficiently to precise transition states. Otherwise, considerable efforts would be spent attempting to
“repair” the “bad” initial guesses.
In contrast to conventional approaches, significantly fewer stochastic population-based methods
have been explored for locating the transition states, making these fertile area for research
investigations. Among the related stochastic optimization techniques proposed to-date, Chaudhury
et al. studied the simulated-annealing approach for locating saddle points [11] and extended their
efforts with the inclusion of gradients information [12]. Bungay et al. considered coupling the
genetic algorithm with eigenvalues information to bias the evolutionary search towards transition
states [13]. Despite their stochastic nature, these methods incidentally also require domain-specific
a priori information for their proper operations due to the vast search space involved. Furthermore,
it is worth noting that while stochastic search methods are able to reliably identify the regions of
interest where transition states may be located, they generally suffer from slow and inefficient
convergence to high-quality solutions.
Taking these cues, we present in this article a novel evolutionary Molecular Memetic Computing
(MMC) methodology requiring zero domain knowledge on the initial guesses of the transition states.
The essence of MMC lies on the tree-based representation of non-cyclic molecules and the covalentbond-driven evolutionary operators in addition to the typical backbone of memetic algorithms—the
population-based global search method and the individual-based life-time learning procedure. In this
article, we employ genetic algorithm [14] for the global search, Berny algorithm [15] for the
individual learning, and make use of the valley-adaptive clearing scheme [16] as the niching strategy
in the spirit of Lamarckian learning [17, 18]. Using the specialized representation and operators, the
MMC is capable of producing offspring that represent meaningful atomic configurations of a
molecule most—if not all—of the time. These in turn shall serve as good initial guesses of the
transition states. Additionally, we formulate fitness function that biases the memetic search towards
finding low-energy transition states. Experiments with five real non-cyclic molecules at the HartreeFock level of ab initio calculation employing the STO-3G basis set have been conducted to assess the
efficacy of the MMC against the recent advances of several state-of-the-art algorithms. Not only
does the MMC uncover the largest set of transition states, but it also incurs the least amount of
computational costs. Furthermore, the evolutionary memetic computing paradigm allows
knowledge discovery that would be valuable for subsequent studies [19].
In the remaining of this article, mathematical formulation of the problem of searching for the
transition states will be briefly discussed in Section 2. Algorithmic details of the MMC will then be
presented in Section 3. Experimental results and in-depth discussions will follow next in Section 4.
These include the presentation of knowledge discovery from the evolutionary data in the forms of
factors affecting the number of transition states, landscape analysis, as well as density distribution of
the transition states. Section 5 finally concludes this article and provides plausible immediate future
works.
2. Problem Formulation
Energy landscape has proven to be a useful conceptual framework in the field of protein folding and
molecular optimization [20–22]. The landscape can be formally defined as an ordered set of three
* |
+ is the set of all possible structural
(
), where
components, i.e.
configurations of a molecule with atoms,
represents the energy terms that serve as the
fitness function, and
denotes the distance measure between two structural
configurations in . The structural configuration space is therefore a set of physically consistent
three-dimensional molecular structures. Each structure
is a vector of
real numbers in
angstroms (Å), representing the Cartesian coordinates of each of the atoms of the molecule. The
potential energy function gives an indication of the height of the energy landscape. Herein, we
work with the first-principle calculations at the Hartree-Fock level using the STO-3G basis set [23]. To
measure the similarity between two structural configurations in , the Ultrafast Shape Recognition
(USR) [24, 25] is considered in this article.
Transition state denotes the configuration where the minimum energy barrier is needed to cross
from the low-energy vicinity of the reactant to the low-energy vicinity of the product. Transition
states play a key role in theoretical and physical chemistry. Their molecular structures and energy
values help in the identification of reaction rates [26] and minimum energy paths [27].
Mathematically, transition state can be defined as a stationary point on the landscape with only
one negative eigenvalue in the Hessian matrix [6], i.e.
* |
( )
+
( ). Scientists are generally interested on
where is an eigenvalue of the Hessian matrix
specific transition states with the following properties:

low-energy, i.e.
( )
where
is some user-defined threshold


| ( )
( )|
( )
unique, i.e.
where and are the maximum
acceptable similarities in the potential energy and the structural configuration spaces,
respectively
precise, i.e. ( )
3. Molecular Memetic Computing
As an emerging field, memetic computing has gained an increasing research attention in the recent
few decades with growing number of successes [28–36]. Inspired by Dawkins’ notion of a meme, it
was first introduced as the Memetic Algorithm (MA)—a marriage between population-based global
search and individual-based local search which respectively facilitate exploration and exploitation of
the search space. Not only MA is capable of escaping local optimum trap, but also it is converging
rapidly to some optimum with sufficient precision due to the life-time learning of the individual. Todate, MAs have been crafted for solving real-world problems in science and engineering more
efficiently [29, 31–32, 37–42].
In this section, specialized representation and operators for the evolution of molecular structure will
be presented. These are crucial for efficient discovery of the low-energy transition states of small
non-cyclic molecules using the molecular memetic computing (MMC) methodology. Specialized
fitness function will also be formulated to enable efficient selection using the stochastic roulettewheel procedure. Finally, the valley-adaptive clearing scheme [16] and Berny algorithm [15] for
maintaining diversity and enabling life-time individual learning, respectively, will be discussed briefly.
3.1. Tree-based Representation
The most straightforward representation of a molecular structure is undoubtedly the string of
concatenated Cartesian coordinates of all its atoms. However, various constraints of a molecular
system, such as covalent bonding, make such representation prone to evolving into nonsensical
structures. To alleviate this problem, we propose a tree-based representation of the molecular
structure where nodes represent atoms, links describes covalent bonds, and subtrees translates into
substructures as depicted in Figure 1. Furthermore, the tree-based representation allows the design
of specialized evolutionary operators that produces meaningful evolved structures most of the time
as will be discussed next.
xC1,yC1,zC1
xN,yN,zN
xH1,yH1,zH1
xH3,yH3,zH3
xH2,yH2,zH2
xH4,yH4,zH4
xC2,yC2,zC2
xH5,yH5,zH5
xH6,yH6,zH6
xC3,yC3,zC3
xH7,yH7,zH7
xH8,yH8,zH8
xC4,yC4,zC4
xO1,yO1,zO1
xO2,yO2,zO2
xH9,yH9,zH9
Figure 1. Tree-based Representation of Non-cyclic Molecule
3.2. Covalent-bond-driven Molecular Evolutionary Operators
3.2.1. Initialization
Constructing the tree-based representation given a seed molecular structure using the procedure
described above produces only one individual. To initialize a population, random sets of mutation
are applied to the seed individual such that multiple structures with different configurations are
achieved as depicted in Figure 2.
Figure 2. Initial Population of MMC
3.2.2. Crossover
Crossover between two molecular configurations aims at interchanging the substructures of
different molecular configurations such that favourable substructures are allowed to replicate across
generations. With reference to the tree-based representation, crossover can be easily performed by
choosing a random link as the crossover point following the alignment of the two molecular
configurations using Kabsch procedure [43] as illustrated in Figure 3. The resulted structures will
therefore most likely observe the constraints imposed by the covalent bonds. Even when the
constraints are not very strictly observed, the evolved structures could have been easily repaired.
chromosomes aligned
Figure 3. Covalent-bond-driven Crossover
3.2.3. Mutation
Mutation of a molecular configuration aims at effecting random changes to covalent bonds about
which rotation and along which stretching may be performed. With reference to the tree-based
representation, mutation can be easily attained by carrying out random rotation about or translation
(subject to minimum and maximum allowable bond lengths [44]) along a randomly chosen link as
shown in Figure 4. In this manner, the resulted structures will always observe the constraints
imposed by the covalent bonds.
(a) bond rotation
(b) bond stretching
Figure 4. Covalent-bond-driven Mutation
3.3. Transition State Fitness Function
The fitness function
is designed to evaluate quantitatively how suitable a generated structural
configuration of a molecule satisfies as a candidate solution of a low-energy transition state. It is
defined mathematically as
( )
( )
(
( )
)(|| |
|
)
where | | and denote the number of negative eigenvalues and a significant small value to prevent
division-by-zero error, respectively. The numerator serves to favour solutions that meet the userdefined low-energy threshold of
. The denominator, on the other hand, is designed to bias the
search towards transition states.
3.4. Valley-adaptive Clearing Scheme
To maintain diversity of the evolving populations, a valley-adaptive clearing scheme [16] is
incorporated to adapt to the non-uniform width of valleys in the energy landscape. Individuals in the
population are clustered into groups that share common valleys. The least fit individuals from each
group are then cleared by relocating them to random regions on the search space. The elite
individual per valley group will then be refined using Berny algorithm [15] described next in the spirit
of Lamarckian learning [17, 18].
3.5. Berny Algorithm
With its capability of converging to the transition states precisely, Berny algorithm enables life-time
individual learning of the elite individual in each valley group. Berny geometry optimization
algorithm is a modified version of the Schlegel algorithm [15] that operates by adjusting the signs of
inadequate eigenvalues. When multiple negative eigenvalues are encountered, all negative
eigenvalues are replaced by their absolute value except the one with the smallest magnitude. When
no negative eigenvalue is present, the sign of the least positive eigenvalue is negated. Following the
life-time individual learning, valley elites that satisfy as transition states are archived and checked
against possible duplicates.
4. Results and Discussions
To showcase the efficacy of the evolutionary MMC methodology detailed above, an experimental
study was performed at the Hartree-Fock level of ab initio calculation of the energy, derivatives, and
eigenvalues with STO-3G basis set. A population size of 10 individuals and an evolution length of
1,000 generations were assumed. As the molecules of interest are the following small non-cyclic
compounds (also shown in Figure 5):





GABA (gamma-aminobutyric acid): a chief inhibitory neurotransmitter that regulates
neuronal excitability in the mammalian central nervous system.
GABOB (gamma-amino-beta-hydroxybutyric acid or buxamin): a derivative of the
neurotransmitter GABA that works mainly as the anti-epileptic drug, offering significant destressing and anti-aging benefits.
Leucine: an essential amino acid important for haemoglobin formation that potently
activates the mammalian target of rapamycin kinase for cell growth regulation.
Tromethamine: an alkalizing agent and buffer in enzymatic reactions that affects the
balance of water and electrolytes in the body; also used in the synthesis of surface-active
agents and pharmaceuticals, in particular, the treatment of metabolic acidosis.
Valine: another essential amino acid characterized with stimulant activity that regulates the
immune function of the body and helps in the development and growth of the muscle as
well as tissue repair; also a main precursor in the penicillin biosynthetic pathway; has been
found useful in treatments involving muscle, mental, and emotional upsets and for insomnia
and nervousness.
(a) GABA
(c) Leucine
(b) GABOB
(d) Tromethamine
(e) Valine
Figure 5. Structural View of the Small Non-cyclic Molecules of Interest
Carbon atoms are represented in green, Nitrogen in blue, Oxygen in red, and Hydrogen in silver.
Figure 6 depicts the number of transition states discovered and the computational cost incurred by
different algorithms, namely the Stochastic Multi-start Local Search using Berny algorithm (SMLS)
[45] as the representative of single-ended methods, the GRRM [46] as the representative of doubleended methods, the Sequential-Niching Memetic Algorithm (SNMA) [30] that represents the recent
advances in niching and memetic algorithms, and finally the proposed MMC methodology with
specialized representation and evolutionary operators. As a double-ended method, the GRRM
requires the reactant and product information to uncover the transition states between them. In
contrast, the SMLS, the SNMA, and the MMC only need to provide initial guesses of the transition
states prior to executing the life-time individual learning procedure. Intuitively, the GRRM shall incur
less computational costs than the other three methods. However, it is witnessed from Figure 6 that
the GRRM is only more efficient than the SMLS and the SNMA. Without the tree-based
representation, initial guesses produced by the SMLS and the SNMA can easily be nonsensical
configurations that require considerable efforts to repair. With tree-based representation and
specialized evolutionary operators, the MMC is observed to be not only more efficient, but also
more effective than the GRRM as well as the SMLS and the SNMA in uncovering the transition states.
In term of number of transition states uncovered, the SNMA actually performs comparably to the
GRRM. The use of niching algorithm in SNMA allows selective life-time individual learning which in
turn gives way to a better exploration of the search space than in the case of SMLS. This
improvement is achievable without requiring the reactant and product information as with the case
of GRRM. In MMC, the use of valley-adaptive clearing scheme as the niching algorithm coupled with
the tree-based representation and the covalent-bond-driven evolutionary operators have further
improved the efficacy of using memetic computing methodology in the discovery of unique, lowenergy transition states of small, non-cyclic molecules.
Figure 6. Performance of SMLS, SNMA, GRMM, and MMC on GABA, GABOB, Leucine, Tromethamine, and Valine in
Terms of Number of Transition States Uncovered and Number of Energy Calculations Performed
With the significantly larger number of uncovered transition states, the use of evolutionary MMC
has furthermore allowed knowledge discovery from data generated over the course of the evolution
rather than requiring prior knowledge as with the case of the other methods. Table 1 tabulates the
characteristics of the uncovered transition states in terms of number of oxygen, carbon, and
hydrogen atoms as well as number of the total atoms and the rotatable bonds. Observation suggests
oxygen atoms play a key role in the flexibility of the molecule, allowing larger number of transition
states to be uncovered. Carbon atoms, in contrast, seem to limit the flexibility of the molecule,
decreasing the number of transition states to be uncovered. Meanwhile, hydrogen atoms are
observed not to play important role in the number of transition states uncovered. Additionally, it is
observed that the number of rotatable bonds play an obvious role in the number of transition states
uncovered while the total number of atoms has not shown any significant impact. The large number
of transition states of GABOB and thromethamine indicates the flexibility of the molecules, allowing
them to interact with broader range of molecular systems. GABOB as an anti-epileptic drug, for
example, is a more effective neuro-inhibitor than GABA from which it is derived. The flexibility that
GABOB can adopt allows it to interact more efficiently with larger set of GABA-ergic system
components in the central nervous systems.
Table 1. Characteristics of Uncovered Transition States in Terms of Number of Atoms, Number of Rotatable Bonds, and
Fitness-Distance Correlation
#transition_state
#oxygen
#carbon
#hydrogen
#total_atoms
#rotatable_bonds
FDC
GABA
256
2
4
9
16
3
26.57
GABOB
643
3
4
9
17
3
3.37
Leucine
305
2
6
13
22
3
54.48
Thrometamine
519
3
4
11
19
3
31.83
Valine
159
2
5
11
19
2
92.20
In Table 1, characteristics of the energy landscape of the molecules are also tabulated as FitnessDistance Correlation (FDC) [13] to measure the difficulty of the landscape. FDC is essentially the
Pearson product moment correlation between the energy differences and the structural differences
of the solutions to the lowest-energy transition states as shown below
(
(
) (
)
)
(
) and ( ) are the covariance and the standard deviation, respectively, while
where
and
represent the energy difference and the USR structure dissimilarity between each transition
state and the lowest-energy transition state, respectively. The FDC categorizes energy landscapes
into well-ordered, rough, or deceptive. High correlation with FDC value above 60 indicates wellordered landscape such that optimization methods can locate the transition states quite easily
whereas low correlation indicates rough landscape in which optimization techniques could be misled
to sub-optimal solutions. Negative correlation, meanwhile, indicates a deceptive landscape where
the lowest-energy transition state is located among the high energy transition states. From Table 1,
the non-cyclic molecules we investigate, except valine, are found to possess rough landscapes. Of
remarkable difficulty is the landscape of GABOB with FDC value of 3.37. The MMC has however
successfully uncovered around six to eight times more of its transition states than those found by
the other methods, owing the success to the tree-based representation and the covalent-bonddriven evolutionary operators. This further showcases the strengths of the evolutionary memetic
computing paradigm when coupled with the appropriate representation and operators such that
little or no effort would be spent exploring the region where no solution may be located.
With the large number of transition states uncovered, we further our analysis to include the
distribution of uncovered transition states by the MMC across the relative energy and the structure
dissimilarity, making reference to the lowest-energy transition states, as shown in Figure 7.
Distribution of the uncovered transition states of the molecules under investigation are depicted in
separate subgraphs using the scatter plots. Of remarkable achievement is the broad range of
transition states uncovered by the MMC as can be concluded from observing the x-axis of the scatter
plots. In other words, the MMC has been capable of discovering unique transition states with high
degree of structural dissimilarities. Observing the y-axis, on the other hand, it is noted that the
uncovered transition states are distinguishable into low- and high-energy transition states. The
presence of high-energy transition states in the region of low dissimilarities could easily hinder the
discovery of low-energy transition states in the region of high dissimilarities. Fortunately, this is not
the case with the MMC. The valley-adaptive clearing scheme used to maintain the population
diversity has indeed been an effective measure for the successful and efficacious exploration of the
energy landscape. Together with the tree-based representation and the covalent-bond-driven
evolutionary operators, the MMC has incurred the least amount of computational costs in addition
to uncovering the largest set of transition states. This shows the evolutionary memetic computing
paradigm is indeed a powerful and efficient tool for solving complex problems when equipped with
appropriate representation and operators.
(a) GABA
(c) Leucine
(b) GABOB
(d) Thrometamine
(e) Valine
Figure 7. Distribution of Uncovered Transition States across Relative Energy and Structure Dissimilarity with reference to
the Lowest-energy Transition State
5. Conclusion and Future Works
With the importance of small molecules in biology, chemistry, and medicine, this article has been
utterly dedicated to the discovery of the transition states of small non-cyclic molecules. The
mathematical formulation and a brief description of the nonlinear programming problem of finding
the transition states have been presented. A novel Molecular Memetic Computing (MMC)
methodology that requires no prior knowledge for the discovery of transition states has been
proposed. At the heart of the method are the tree-based representation of non-cyclic molecules and
the covalent-bond-driven evolutionary operators. In the spirit of Lamarckian learning [17, 18], the
specialized representation and operators are used to complement the memetic computing
technique comprising a genetic algorithm [14] for the population-based global search, the Berny
algorithm [15] for the life-time individual learning, and the valley-adaptive clearing scheme [16] for
maintaining the population diversity.
The evolutionary MMC methodology shows promising application of memetic computing techniques
in the field of molecular optimization by demonstrating excellent efficacy against several state-ofthe-art approaches. Not only did the MMC uncover the largest number of transition states, but it
also incurred the least amount of computational costs. The use of evolutionary MMC, furthermore,
allows knowledge discovery from data generated over the course of the evolution rather than
requiring prior knowledge as with the case of many earlier methods [6–13]. It has been found out
that oxygen atoms play a key role in the flexibility of the molecule, allowing larger number of
transition states to be uncovered. In contrast, carbon atoms have been observed to limit the
flexibility of the molecule, decreasing the number of transition states to be uncovered. Meanwhile,
hydrogen atoms have not been observed to play any important role in the number of transition
states uncovered. Additionally, it has also been found out—as one would expected—that the
number of rotatable bonds play an obvious role in the number of transition states uncovered while
the total number of atoms has not shown any significant impact. On a side note, large number of the
transition states of GABOB and thromethamine suggests the flexibility of the molecules, allowing
them to interact with broader range of molecular systems. GABOB as an anti-epileptic drug, for
instance, is a more effective neuro-inhibitor than GABA, from which it is derived. The flexibility that
GABOB can adopt allows it to interact more efficiently with larger set of GABA-ergic system
components in the central nervous systems.
With the generic tree-based representation, the MMC can be easily adapted for subsequent studies
of other small non-cyclic molecules that are important in biology, chemistry, and medicine. To cover
molecules that involve ring structures, an extension to the tree-based representation shall be
studied. Extensions to the current covalent-bond-driven evolutionary operators shall also be
devised. To allow working with larger molecules, techniques to intelligently restrict the execution of
individual learning procedure [47–50] shall be explored. Finally, Web Service that would allow
scholars to search for the transition states of molecular systems of their choices shall be developed.
In the end, our aim shall be to paves the way to establish the landscape connectivity graph, the
master piece of the comprehensive chemical reactions.
References
1. Chen, J., S.J. Swamidass, Y. Dou, J. Bruand, and P. Baldi, “ChemDB: A Public Database of
Small Molecules and Related Chemoinformatics Resources,” Bioinformatics, 21(22),
pp.4133–4139, 2005.
2. Lagorce, D., T. Pencheva, B.O. Villoutreix, and M.A. Miteva, “DG-AMMOS: A New Tool to
Generate 3D Conformation of Small Molecules Using Distance Geometry and Automated
Molecular Mechanics Optimization for in silico Screening,” BMC Chemical Biology, 9(1),
2009.
3. Sperandio, O., M. Souaille, F. Delfaud, M.A. Miteva, and B.O.Villoutreix, “MED-3DMC: A New
Tool to Generate 3D Conformation Ensembles of Small Molecules with A Monte Carlo
Sampling of the Conformational Space,” European Journal of Medicinal Chemistry, 44(4),
pp.1405–1409, 2009.
4. Miteva, M.A., F. Guyon, and P. Tufféry, “Frog2: Efficient 3D Conformation Ensemble
Generator for Small Compounds,” Nucleic Acids Research, 38, pp.W622–W627, 2010.
5. Ellabaan, M.M.H., Y.S. Ong, M.H. Lim, and J.L. Kuo, “Finding Multiple First-order Saddle
Points Using A Valley-adaptive Clearing Genetic Algorithm,” in Proceedings of 2009 IEEE
International Symposium on Computational Intelligence in Robotics and Automation,
pp.457–462, 2009.
6. Trygubenko, S.A. and D.J. Wales, “A Doubly Nudged Elastic Band Method for Finding
Transition States,” Journal of Chemical Physics, 120(5), pp.2082–2094, 2004.
7. Carr, J.M., S.A. Trygubenko, and D.J. Wales, “Finding Pathways Between Distant Local
Minima,” Journal of Chemical Physics, 122(23), pp.234903–234909, 2005.
8. Koslover, E.S. and D.J. Wales, “Comparison of Double-ended Transition State Search
Methods,” Journal of Chemical Physics, 127(13), pp.134102–134113, 2007.
9. Sheppard, D., R. Terrell, and G. Henkelman, “Optimization Methods for Finding Minimum
Energy Paths,” Journal of Chemical Physics, 128(13), pp.134106–134115, 2008.
10. Olsen, R.A., G.J. Kroes, G. Henkelman, A. Arnaldsson, and H. Jónsson, “Comparison of
Methods for Finding Saddle Points without Knowledge of the Final States,” Journal of
Chemical Physics, 121(20), pp.9776–9792, 2004.
11. Chaudhury, P. and S.P. Bhattacharyya, “A Simulated Annealing Based Technique for Locating
First-order Saddle Points on Multidimensional Surfaces and Constructing Reaction Paths:
Several Model Studies,” Journal of Molecular Structure: THEOCHEM, 429, pp.175–186, 1998.
12. Chaudhury, P., S.P. Bhattacharyya, and W. Quapp, “A Genetic Algorithm Based Technique for
Locating First-order Saddle Point Using A Gradient-dominated Recipe,” Chemical Physics,
253(2–3), pp.295–303, 2000.
13. Bungay, S.D., R.A. Poirier, and R.J. Charron, “Optimization of Transition State Structures
Using Genetic Algorithms,” Journal of Mathematical Chemistry, 28(4), pp.389–401, 2000.
14. Holland, J.H., Adaptation in Natural and Artificial Systems, MIT Press, 1992.
15. Schlegel, H.B., “Optimization of Equilibrium Geometries and Transition Structures,” Journal
of Computational Chemistry, 3(2), pp.214–218. 1982.
16. Ellabaan, M.M.H. and Y.S. Ong, “Valley-adaptive Clearing Scheme for Multimodal
Optimization Evolutionary Search,” in Proceedings of the Ninth International Conference on
Intelligent Systems Design and Applications, pp.1–6, 2009.
17. Ong, Y.S. and A.J. Keane, “Meta-lamarckian Learning in Memetic Algorithm,” IEEE
Transactions on Evolutionary Computation, 8(2), pp.99–110, 2004.
18. Le, M.N., Y.S. Ong, Y. Jin, and B. Sendhoff, “Lamarckian Memetic Algorithms: Local Optimum
and Connectivity Structure Analysis,” Memetic Computing, 1(3), pp.175–190, 2009.
19. Ang, J.H., K.C. Tan, and A.A. Mamun, “An Evolutionary Memetic Algorithm for Rule
Extraction,” Expert Systems with Applications, 37(2), pp.1302–1315, 2010.
20. Brooks, C.L. 3rd, J.N. Onuchic, and D.J. Wales, “Statistical Thermodynamics. Taking A Walk on
A Landscape,” Science, 293(5530), pp.612–613, 2001.
21. Wales, D.J., Energy Landscapes: Applications to Clusters, Biomolecules and Glasses,
Cambridge University Press, 2003.
22. Soh, H., Y.S. Ong, Q.C. Nguyen, Q.H. Nguyen, M.S. Habibullah, T. Hung, and J.L. Kuo,
“Discovering Unique, Low-Energy Pure Water Isomers: Memetic Exploration, Optimization,
and Landscape Analysis,” IEEE Transactions on Evolutionary Computation, 14(3), pp.419–
437, 2010.
23. Jensen, F., Introduction to Computational Chemistry, Wiley, 1999.
24. Zhou, T., K. Lafleur, A. Caflisch, “Complementing Ultrafast Shape Recognition with An Optical
Isomerism Descriptor,” Journal of Molecular Graphics and Modelling, 29(3), pp.443–449,
2010.
25. Ballester, P.J. “Ultrafast Shape Recognition: Method and Applications,” Future Medicinal
Chemistry, 3(1), pp.65–78, 2011.
26. Truong, T.N. and D.G. Truhlar “Ab initio Transition State Theory Calculations of the Reaction
Rate for OH+CH4→H2O+CH3,” Journal of Chemical Physics, 93(3), pp.1761–1769, 1990.
27. Henkelman, G. and G. Jóhannesson, “Methods for Finding Saddle Points and Minimum
Energy Paths,” Progress in Theoretical Chemistry and Physics, 5, pp.269–302, 2002.
28. Moscato, P., “On Evolution, Search, Optimization, Genetic Algorithms, and Martial Arts:
Towards Memetic Algorithms,” Caltech Concurrent Computation Program, 1989.
29. Neri, F., J. Toivanen, G. Cascella, and Y.S. Ong, “An Adaptive Multimeme Algorithm for
Designing HIV Multidrug Therapies,” IEEE/ACM Transactions on Computational Biology and
Bioinformatics, 4(2), pp.264–278, 2007.
30. Vitela, J.E. and O. Castanos, “A Real-coded Niching Memetic Algorithm for Continuous
Multimodal Function Optimization,” in Proceedings of 2008 IEEE Congress on Evolutionary
Computation, pp.2170–2177, 2008.
31. Ong, Y.S., M.H. Lim, and X.S. Chen, “Memetic Computation—Past, Present, & Future,” IEEE
Computational Intelligence Magazine, 5(2), pp.24–36, 2010.
32. Chen, X.S., Y.S. Ong, M.H. Lim, and K.C. Tan, “A Multi-facet Survey on Memetic
Computation,” IEEE Transactions on Evolutionary Computation, 15(5), pp.591–607, 2011.
33. Cheng, H.C., T.C. Chiang, and L.C. Fu, “A Two-stage Hybrid Memetic Algorithm for
Multiobjective Job Shop Scheduling,” Expert Systems with Applications, 38(9), pp.10983–
10998, 2011.
34. Zhang, J., Z. Zhan, Y. Lin, N. Chen, Y. Gong, J. Zhong, H.S.H. Chung, Y. Li, and Y. Shi,
"Evolutionary Computation Meets Machine Learning: A Survey," IEEE Computational
Intelligence Magazine, 6(4), pp.68–75, 2011.
35. Jin, Y., K. Tang, X. Yu, B. Sendhoff, and X. Yao, “A Framework for Finding Robust Optimal
Solutions Over Time,” Memetic Computing, in Press.
36. Le, M.N., Y.S. Ong, S. Menzel, Y. Jin, and B. Sendhoff, "Evolution by Adapting Surrogates,"
Evolutionary Computation, in Press.
37. Chia, J.Y., C.K. Goh, K.C. Tan, and V.A. Shim, "Memetic Informed Evolutionary Optimization
via Data Mining," Memetic Computing, 3(2), pp.73–87, 2011.
38. Damas, ., O. Cord n, and . antamar a, edi a Ima e Re istration sin
o tionary
Computation: An Experimental Survey," IEEE Computational Intelligence Magazine, 6(4),
pp.26–42, 2011.
39. Meng, Y., Y. Zheng, and Y. Jin, "Autonomous Self-reconfiguration of Modular Robots by
Evolving A Hierarchical Mechanochemical Model," IEEE Computational Intelligence
Magazine, 6(1), pp.43–54, 2011.
40. Shim, V.A., K.C. Tan, J.Y. Chia, and J.K. Chong, "Evolutionary Algorithms for Solving Multiobjective Travelling Salesman Problems," Flexible Services and Manufacturing, 23(2),
pp.207–241, 2011.
41. Le, M.N., Y.S. Ong, Y. Jin, and B. Sendhoff, "Gene Meets Meme in Computational
Intelligence: A Theoretic Modeling of Symbiotic Evolution," IEEE Computational Intelligence
Magazine, 7(1), pp.20–35, 2012.
42. Thomas S.A. and Y. Jin, "Evolving Connectivity Between Genetic Oscillators and Switches
Using Evolutionary Algorithms," Journal of Bioinformatics and Computational Biology, in
Press.
43. Kabsch, W., “A Solution for the Best Rotation to Relate Two Sets of Vectors,” Acta
Crystallographica, 32(5), pp.922–923, 1976.
44. Weast, R.C., Handbook of Chemistry and Physics, CRC Press, 1984.
45. Kok, S. and C. Sandrock, “Locating and Characterizing the Stationary Points of the Extended
Rosenbrock Function,” Evolutionary Computation, 17(3), pp.437–453, 2009.
46. Maeda, S., Y. Matsuda, S. Mizutani, A. Fujii, and K. Ohno, “Long-range Migration of A Water
Molecule to Catalyze A Tautomerization in Photoionization of the Hydrated Formamide
Cluster,” Journal of Physical Chemistry A, 114(44), pp.11896–11899, 2010.
47. Handoko, S.D., C.K. Kwoh, and Y.S. Ong, “Using Classification for Constrained Memetic
Algorithm: A New Paradigm,” in Proceedings of IEEE International Conference on Systems,
Man and Cybernetics, pp.547–552, 2008.
48. Handoko, S.D., C.K. Kwoh, and Y.S. Ong, “Classification-assisted Memetic Algorithms for
Equality-constrained Optimization Problems,” Lecture Notes in Computer Science, 5866,
pp.391–400, 2009.
49. Handoko, S.D., C.K. Kwoh, and Y.S. Ong, “Feasibility Structure Modeling: An Effective
Chaperone for Constrained Memetic Algorithms,” IEEE Transactions on Evolutionary
Computation, 14(5), pp.740–758, 2010.
50. Handoko, S.D., C.K. Kwoh, Y.S. Ong, and J. Chan, “Classification-assisted Memetic Algorithms
for Solving Optimization Problems with Restricted Equality Constraint Function Mapping,” in
Proceedings of 2011 IEEE Congress on Evolutionary Computation, pp.1209–1216, 2011.