A simplex nelder mead genetic algorithm for minimizing molecular potential energy function
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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.
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LOGO
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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14](https://image.slidesharecdn.com/asimplexnelder-meadgeneticalgorithmforminimizingmolecularpotentialenergyfunction-131022141052-phpapp02/85/A-simplex-nelder-mead-genetic-algorithm-for-minimizing-molecular-potential-energy-function-14-320.jpg)
A simplex nelder mead genetic algorithm for minimizing molecular potential energy function
- 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 Company LOGO
- 2. Company LOGO Scientific Research Group in Egypt www.egyptscience.net
- 3. Company LOGO Agenda 1. Motivations 2. Introduction 3. Problem definition 4. The proposed GNMA 5. Performance analysis 6. Numerical experiments 7. Conclusions
- 4. Company LOGO 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.
- 5. Company LOGO 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 5
- 6. Company Introduction LOGO 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. 6 6
- 8. Company LOGO Introduction NMR (Nuclear Magnetic Resonance) but this is very expensive and time-consuming 8 8
- 9. Company LOGO 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 9 9 9
- 10. Company LOGO Problem definition The molecular model considered here consists of a chain of m atoms centered at x1, . . . , xm, in a 3-dimensional space 10 10 10
- 11. Company LOGO 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. 11 11 11
- 12. Company LOGO 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 . 12 12 12
- 13. Company LOGO 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 . 13 13 13
- 14. Company LOGO 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). 14 14 14
- 15. Company LOGO 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. 15 15 15
- 16. Company LOGO 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 . 16 16
- 17. Company LOGO 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. 17 17 17
- 18. Company LOGO 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 18 18 18
- 19. Company LOGO The proposed algorithm (GNMA) • GNMA uses arithmetical crossover 19 19 19
- 21. Company LOGO The Nelder-Mead algorithm (Cont.) 21 21 21
- 22. Company LOGO The Nelder-Mead algorithm (Cont.) 22 22 22
- 23. Company LOGO The Nelder-Mead algorithm (Cont.) 23 23 23
- 24. Company LOGO The Nelder-Mead search strategy 24 24 24
- 26. Company LOGO The proposed GNMA (Cont.) 26 26 26
- 27. Company LOGO Performance analysis The main contributions in GNMA lies in two components • Population partitioning •Final intensification(exploitation) by applying NelderMead algorithm in the elite solution. 27 27 27
- 28. Company LOGO Performance analysis(The efficiency of population partitioning) 28 28 28
- 29. Company LOGO Performance analysis(The efficiency of Nelder-Mead Algorithm) 29 29 29
- 30. Company LOGO 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. 30 30 30
- 35. Company LOGO 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. 35 35 35
- 36. Company LOGO Thank you [email protected] http://www.egyptscience.net Authors Ahmed Fouad Ali, Aboul Ella Hassanien