The document describes a new genetic Nelder-Mead algorithm (GNMA) for minimizing molecular potential energy functions. GNMA uses a population partitioning approach and applies genetic operations to submatrices before using Nelder-Mead optimization on elite solutions. It was tested on a simplified molecular potential energy function and compared to 9 other algorithms, showing promising performance with less computational expense than other methods. The authors conclude GNMA is an efficient approach for molecular structure optimization problems.

1. Scientific Research Group in Egypt (SRGE)
A simplex Nelder-Mead genetic algorithm for
minimizing molecular potential energy function
Ahmed Fouad Ali and Aboul Ella Hassanien
Scientific Research Group in Egypt (SRGE)
http://www.egyptscience.net
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Scientific Research Group in Egypt
www.egyptscience.net

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Agenda
1. Motivations
2. Introduction
3. Problem definition
4. The proposed GNMA
5. Performance analysis
6. Numerical experiments
7. Conclusions

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Motivations
•The determination of the three-dimensional structure of a molecule
can be formulated as a continuous global minimization problem.
• The problem is that the number of local minimizers of this
function grows exponentially with the molecule size.
• Many optimization methods can be applied to this problem, such
as branch and bound methods, smoothing methods, simulated
annealing, genetic algorithms.
•To explore the capability of the proposed algorithm we use a
scalable simplified molecular potential energy function with well
known properties.

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Introduction
A Protein is a chain of amino acids, also referred to as residues.
R
NH3+ C COOAmino group
H
Carboxylic
acid group
Different side
chains, R,
determine the
properties of 20
amino acids.
An amino acid
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Gly
Leu
a-helix
Ser
Pro
b-sheet
Proteins consist of a long chain
of amino acids called the
primary structure.
The constituent amino acids may
encourage hydrogen bonding and
form regular structures, called
secondary structures.
The secondary structures fold
together to form a compact
3-dimensional or tertiary
structure.
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Introduction
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Introduction
NMR (Nuclear Magnetic Resonance)
but this is very expensive and time-consuming
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Introduction
The problem can be formulated as a global
minimization problem, as it is assumed that the
tertiary structure occurs at the global minimum of the
free energy function of the primary sequence
Tertiary structure is
believed to minimize potential energy:
Min V (x)
where x = atom coordinates
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Problem definition
The molecular model considered here consists of a chain of
m atoms centered at x1, . . . , xm, in a 3-dimensional space
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Problem definition
The force field potentials corresponding to bond lengths,
bond angles, and torsion angles are defined respectively.
Where c1ij is the bond stretching force constant,
c2ij is the angle bending force constant,
c3ij is the torsion force constant.
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Problem definition
There is a potential E4 which characterizes the 2-body
interaction between every pair of atoms separated by
more than two covalent bonds along the chain.
where rij is the Euclidean distance between atoms xi and xj .
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Problem definition
In most molecular conformational predictions, all covalent
bond lengths and covalent bond angles are assumed to be
fixed at their equilibrium values r0ij and θ0ij , respectively.
The molecular potential energy function reduces to
E3 + E4.
where rij is the Euclidean distance between atoms xi and xj .
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Problem definition
• From Eq. 3 and Eq. 4, the expression for the potential
energy as a function of the torsion angles takes the form
• The problem is then to find ω14, ω25, . . . , ω(m−3)m,
considering ωij ∈ [0, 5], which corresponds to the global
minimum of the function E, represented by Eq.(5).
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Problem definition
Finally, the function f(x) can defined as
Despite these simplification, the problem remains very
difficult. A molecule with as few as 30 atoms has 227 =
134, 217, 728 local minimizers.
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The proposed algorithm (GNMA)
• We propose a new Genetic Algorithm (GA) based
method, called Genetic Nelder-Mead Algorithm
(GNMA)
• A population of chromosomes are coded in a one big
matrix.
• This matrix is partitioned into several sub-matrices.
• The genetic operations are applied on the partitioned
sub-matrices.
• In the last stage, an exploitation process starts to
refine the best candidates obtained by applying Nelder
– Mead Algorithm .
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The proposed algorithm (GNMA)
• GNMA starts with an initial population P0 of size μ which is
coded into a one big matrix PM0 .
• In this matrix a row is representing a chromosome and
each column shows the values of the corresponding
gene in all chromosomes.
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The proposed algorithm (GNMA)
• The general population matrix Pt at generation is given by
Pt is partitioned into υ x η sub-matrices Pt(i;j) , i = 1,…, υ, j =
1,…, η, then the crossover and mutation operations are
applied to update each sub-population
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The proposed algorithm (GNMA)
• GNMA uses arithmetical crossover
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The Nelder-Mead algorithm
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The Nelder-Mead algorithm (Cont.)
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The Nelder-Mead algorithm (Cont.)
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The Nelder-Mead algorithm (Cont.)
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The Nelder-Mead search strategy
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The proposed GNMA
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The proposed GNMA (Cont.)
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Performance analysis
The main contributions in GNMA lies in two components
• Population partitioning
•Final intensification(exploitation) by applying NelderMead algorithm in the elite solution.
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Performance analysis(The efficiency of
population partitioning)
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Performance analysis(The efficiency of
Nelder-Mead Algorithm)
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Numerical experiments
•The GNMA is compared with 9 Methods
•Four methods are based on two real coded crossover operators
Weibull crossover WX and LX and two mutation operators LLM and
PM. WX-PM, WX-LLM, LX-LLM , LX-PM .
•Variable neighborhood search based method (VNS-3), (VNS-123)
•Genetic algorithm (GA)
• (rHYB) method denotes the staged hybrid GA with a reduced
simplex and a fixed limit for simplex iterations
• (qPSO) method is a hybrid particle swarm optimization (PSO) in
which quadratic approximation operator is hybridized with PSO.
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Numerical experiments
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Numerical experiments
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Numerical experiments
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Numerical experiments
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Conclusion
In this paper, a new genetic Nelder-Mead based algorithm,
called GNMA, has been proposed to minimize the molecular
potential energy function.
The use of the partitioning process effectively assists an
algorithm to achieve promising performance and wide
exploration before stopping the search.
The Nelder-Mead algorithm has been inlaid in GNMA to
accelerate the search process and achieve deep exploitation with
the best individual before the end of the algorithm.
The comparison with 9 benchmark methods have been
presented to show the efficiency of GNMA.
The compared results indicate that GNMA is promising and
it is less expensive and much cheaper than other methods.
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Thank you
Ahmed_fouad@ci.suez.edu.eg
http://www.egyptscience.net
Authors Ahmed Fouad Ali, Aboul Ella Hassanien