rithm and Particle Swarm Optimization - Semantic Scholar

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International Journal of Fuzzy Systems, Vol. 13, No. 2, June 2011

Protein 3D HP Model Folding Simulation Using a Hybrid of Genetic Algorithm and Particle Swarm Optimization Cheng-Jian Lin and Shih-Chieh Su Abstract1 Given the amino-acid sequence of a protein, the prediction of a protein’s tertiary structure is known as the protein folding problem. The protein folding problem in the hydrophobic-hydrophilic lattice model is the problem of finding the lowest energy conformation. This is the NP-complete problem. In order to enhance the procedure performance for predicting protein structures, a hybrid genetic-based particle swarm optimization (PSO) is proposed. Simulation results indicate that our approach outperforms the existing evolutionary algorithms. The method can be applied successfully to the protein folding problem based on the three-dimensional hydrophobic-hydrophilic lattice model. Keywords: Protein structure prediction, three- dimensional HP lattice model, particle swarm optimization, genetic algorithm.

1. Introduction The prediction of protein structure from its amino-acid sequence is one of the most prominent problems in computational biology. A protein’s function depends mainly on its tertiary structure, which in turn depends on its primary structure. Folding mistakes will create proteins with abnormal shapes which are the main causes of numerous diseases, such as cystic fibrosis, Alzheimer’s, and mad cow. If it were possible to predict proteins’ tertiary structures of from their sequences with high accuracy, it would be able to treat these diseases better. The knowledge of protein tertiary structures also has other applications, such as in the structure-based drug design field [1]. Currently, protein structures are primarily determined by techniques such as MRI (magnetic resonance imaging) and X-ray crystallography, which are expensive in terms of equipment, computation, and time. Corresponding Author: C.-J. Lin is with the Department of Computer Science and Information Engineering, National Chin-Yi University of Technology, No. 35, Lane 215, Sec. 1, Chung-Shan Rd., Taiping City, Taichung County, Taiwan, 411 E-mail: [email protected] Manuscript received 11 Jan. 2010; revised 15 Sept. 2010; accepted 27 Dec. 2010.

Additionally, these techniques require isolation, purification, and crystallization of the target protein. Computational approaches to protein structure prediction are therefore very attractive. The difficulties in solving protein structure prediction problems stems from two major challenges: (1) finding good measures to verify the qualities of candidate structures; and (2) given such measures, determining optimal or close-to-optimal structures for a given amino-acid sequence [2]. The Hydrophobic-Polar model (HP model) is a simplified model which has become very popular. The HP model is one of the widely used models. For the 2D HP model, a two-dimensional square lattice is widely used [1]-[12]. The real application of evolutionary algorithms [2]-[4] to the two-dimensional HP model was found to be impracticable. It has been claimed that the characterization completely depends on the accuracy of identifying a meaningful structure. The biological activity of a protein depends on its three-dimensional (tertiary) structure. Thus, to determine that structure has become one of the most central problems in bioinformatics. The 3D HP model generally adopts a three dimensional cubic lattice. It is observed that the characterization completely depends on the accuracy of identifying the data points which are meaningful [13]. Fortunately, the three-dimensional HP model is similar to the real protein structure. Recently, many researchers [2],[14]-[18] have used evolutionary algorithms, such as the genetic algorithms (GA), for solving the protein folding problem. The Genetic Algorithm (GA) is a method of optimizing random searching by multiple pathways. In order to replace fixed rules with random calculation in the searches for proper conformation, the GA is chosen as the solution in the PSP problem in our study. In general, the GA gives very robust performance due to its inherent advantages; however, its efficiency in solving the PSP still needs further improvements. There are two key operations in the GA, i.e., crossover and mutation. Crossover is capable of linking two different chromosomes very efficiently to form a new conformation, while mutation will not be restricted by local optimum. The limits of the HP model, on the other hand, will generate many invalid conformations after operations of crossover and mutation, which results in high reduction of the predication efficiency. Therefore, some researchers have proposed various hy-

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C.-J. Lin et al.: Protein 3D HP Model Folding Simulation Using a Hybrid of Genetic Algorithm and Particle Swarm Optimization

brid methods aiming to improve the efficiency of GA recently. The key-point to improve GA lies primarily in operating optimization of crossover and mutation, while relative methods have been developed in this direction. For example, the particle swarm optimization (PSO) has been chosen to select better mutation in genes. Lin et al. [19]-[20] has achieved good results in engineering and computing science by applying the PSO to optimize the mutation process. In this paper, we focus on improving the three-dimensional hydrophobic-polar (HP) lattice model [18] and propose a hybrid genetic-based particle swarm optimization (HGA-PSO) to tackle the PSP problem in the 3D HP model. The simulation results indicate that our proposed 3D HP model in protein folding problem can yield much better performance than other current evolutionary algorithms. The remainder of this paper is structured as follows. Section 2 sets out the preliminaries and the formal definition of the protein folding problem in the 3D HP lattice model. Section 3 describes our approach in detail. The hybrid genetic-based particle swarm optimization is presented. Experimental results obtained by our method and by other methods are compared in Section 4. Finally, the conclusion is given in Section 5.

2. Preliminaries A. The 3D HP Protein Folding Problem Lattice proteins are highly simplified computer models of proteins which are used to investigate protein folding. Dill [1] proposes the hydrophobic-polar model. Because proteins are such large molecules, containing hundreds or thousands of atoms, it is not possible with current technology to simulate more than a few microseconds of their behavior in complete atomic detail. Hence, folding of real proteins cannot be simulated in a computer operation. Lattice proteins, however, are simplified in two ways: the amino acids are modeled as single “beads” rather than by every atom, and the beads are restricted to a rigid (usually cubic) lattice. This simplification means they can quickly fold to their energy minima in a time short enough to be simulated. Lattice proteins are made to resemble real proteins by introducing an energy function [9], a set of conditions which specify the energy of interaction between neighboring beads, usually taken to be those occupying adjacent lattice sites. The energy function mimics the interactions, which include hydrophobic and hydrogen bonding effects, between amino acids in real proteins. The beads are divided into types, and the energy function specifies the interactions, depending on the bead type, just as different types of amino acid interact differently.

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Figure 1. An optimal conformation for the sequence “(HP)2PH(HP)2(PH)2HP(PH)2” in 3D lattice model.

B. Calculating The Free Energy One of the most popular lattice models, the HP model, features just two bead types: H (hydrophobic or non-polar) and P (hydrophilic or polar). An instance is shown in Figure 1 for the 3D HP lattice model [6]. The black beads denote the hydrophobic amino acid and white beads denote the hydrophilic. The dotted line denotes the H-H contacts (free energy) in the conformation, which are assigned an energy value of -1. The free energy is minimum; the number of H-H contacts is the maximum. Figure 1 shows a protein structure with 11 H-H contacts (energy= -11). Since the native state of a protein generally corresponds to the lowest free energy state for the protein, the optimal conformation in the HP model is the one that has the maximum number of H-H contacts which gives the lowest energy value. The free energy for the protein can be calculated by using the following formulae [21]:

(1)

(2) where the parameter (3) Hence, the protein folding problem can be transformed into an optimization problem, i.e., to calculate the minimal free energy of the protein folding conformation. As a result, the following problem can be formally defined: given an HP sequence s = s1, s2, ..., sn, find an energy-minimizing conformation of s; that is: find c* ∈ C(s) such that E(c*) = min{E(c) | c ∈ C}, where C(s) is

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the set of all valid conformations for s [22].

3. Methods A. Particle Swarm Optimization The idea of particle swarm optimization was originally introduced by Kennedy and Eberhart in 1995 to study social and cognitive behavior [23]-[24]. The idea originated in studies on the synchronous flocking of birds and the schooling of fish. The PSO has come to be widely used as a problem solving method in engineering and computer science. This algorithm has several highly desirable attributes, including a basic algorithm that is very easy to understand and implement. It is similar in some ways to evolutionary algorithms, but requires less computational bookkeeping and generally fewer lines of code. In the PSO, the trajectory of each individual in the search space is adjusted by dynamically altering the velocity of each particle, according to its own flight experience and the flight experience of the other particles in the search space. The position vector and the velocity vector of the ith particle in the D-dimensional search space can be represented by X i = ( xi1 , xi 2 , xi 3 ,......, xid ) and Vi = (ν i1 ,ν i 2 ,ν i 3 ,......,ν id ) , respectively. According to a user-defined fitness function, suppose that the best position of each particle (which corresponds to the best fitness value obtained by the particle at that time) is Pi = ( pi1 , pi 2 , pi 3 ,......, pid ) and that the fittest particle found so far is Pg = ( p g1 , p g 2 , p g 3 ,......, p gd ) . Then the new velocities and the positions of the particles for the next fitness evaluation are calculated using the following two equations: ν idk +1 = ν idk + c1 × rand (⋅) × ( Pid − xidk ) + c 2 × rand (⋅) × ( Pgd − xidk ) (2)

xidk +1 = xidk + vidk +1

(3)

where c1 and c2 are constants known as acceleration coefficients, and rand (⋅) are two separately generated uniformly distributed random numbers in the range [0,1]. The concept of the updated velocity is illustrated in Figure 2. The first part of Equation (2) represents the previous velocity, which provides the necessary momentum for particles to roam across the search space. The second part of Equation (2), known as the “cognitive” component, represents the personal thinking of each particle. The cognitive component encourages the particles to move toward their own best positions found up to that point. The third part of Eq. (2) is known as the “social” component, which represents the collaborative effect of the particles in finding the global optimal solution. The social component always pulls the particles toward the best global particle found so far. Changing

velocity enables every particle to search around its best individual and global positions. After the new velocity is obtained, the particle updates the position by following Eq. (3). When every particle is updated, the fitness value of each particle is calculated again. If the fitness value of the new particle is higher than those of the local best, then the local best will be replaced with the new particle; and further, if the local best is better than the current global best, then we replace the global best with the local best in the swarm.

Figure 2. The diagram of the updated velocity in the PSO.

B. Particle Swarm Optimization Figure 3 shows the flowchart of the proposed hybrid genetic-based particle swarm optimization (HGA-PSO). The hybrid learning algorithm works with a population of candidate solutions. At each generation, the n best individuals of the population are selected based on their fitness (the minimum free energy). The details are as follows: z Initialization step: If the input amino acid sequence is of length n, then each individual in the population is a string of length n-1 over the symbols = {R, L, F, B, D, U}, and that denotes a valid conformation in the 3D square lattice [10]. The symbols R, L, F, B, D and U are used to denote the fold directions right, left, forward, backward, up and down in the encoding scheme, respectively. An initial population is generated randomly and initializes an n-1 dimensional space within a fixed range. Figure 4 shows that the adopted schemes for representing internal movements are absolute directions. z Reproduction step: Reproduction is a process in which individual strings are copied according to their fitness value. In this study, we use the roulette-wheel selection method [25]: a simulated roulette is spun for this reproduction process. The best performing individuals (that is, the whole of the top half is considered as best) [26] advance to the next generation. The other half is generated to perform crossover and mutation operations on individuals in the top half of the parent generation.


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crossed and separated using a two-point crossover operation. In Figure 5, new individuals are created by exchanging the site’s direction between points i and j (the selected sites of an individual’s parents). After this operation, the individuals with poor performances are replaced by the newly produced offspring.

Figure 5: Two-point crossover operation between the i and j chromosomes.

Figure 3. Flowchart of the proposed hybrid learning algorithm.

Figure 6. Mutation operation using PSO with the ith-jth chromosomes. z

Figure 4. The schemes of the internal coordinates (the black cube represents the current location). z

Crossover Step: Reproduction directs the search toward the best existing individuals but does not create any new individuals. In nature, an offspring has two parents and inherits genes from both. The main operation used when working on the parents is the crossover operation, the operation of which occurred for a selected pair with a crossover rate that was set to 0.8 in this study. The first step is to select the individuals from the population for the crossover. Tournament selection [25] is used to select the top-half of the best performing individuals [26]. The individuals are

Mutation step: When the mutation points are selected, the mutation of an individual is as shown in Figure 6. For the HP protein folding problem, every amino-acid residue owns each folding direction that is unique to the site. GA is incapable of efficient mutation operations aimed at each residue. Therefore, we propose a mutation operation based on particle swarm optimization. Equation (2) represents variations of the folding direction to the next generation. We calculate the difference between the current particle and the local best particle. We also calculate the difference between the current particle and the global best particle. Equation (3) represents variations of the current position that updates the particles. Through the mutation step, only one best child can survive to replace its parent and enter the next generation. This involves PSO changing the local structure between i and j, where i and j are two randomly determined sequence positions, such that 1 ≤ i ≤ j ≤ n (the length of protein sequence). Hence, we employ the merits of PSO to improve the mutation mechanism. In order to avoid entrapment in a local optimal solution and to ensure the searching capability of the near global optimal solution, mutation plays an important role in PSO. It is a recently invented high performance optimizer that possesses sev-

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eral highly desirable attributes, including the fact that the basic algorithm is very easy to understand and implement. It requires less computational memory and fewer lines of code. Each particle has a velocity vector ν and a position vector x to represent a possible solution. There are three possibilities for each particle during evolution: (1) Remain itself; (2) move towards the present optimum; each particle “remembers” its own personal best position that it has found, called the local best; and (3) move towards the best population it has encountered. Each particle is also influenced by the best position found by any particle in the swarm, called the global best. z Local search step: After mutation, a local search is the same as the crossover operation. New individuals are created by changing the site’s direction between point i and point j. Two motions have produced a new direction. The new folding direction is superior to the original direction. One of the two motions is selected by competition. If the new folding direction is not better than the original direction, the original direction will not change. A local search can perform an intensive search for a new and better solution. This is similar to the mutation operation. The local search is different from the mutation operation in terms of the rules. A local search has system rules and effectively finds a local solution. a) Opposite motion: As shown in Figure 7, the motion of a local structure is in the opposite direction. We can change the local structure between two randomly determined sequence positions. All the residue directions are right, down, backward, up, backward, right to left, up, forward, down, forward, left. The inversion method can advance in the opposite direction, which is the direction of repulsion.

the best direction in a local search by three methods. The new folding direction is superior to the original direction. If the new folding direction is not better than the original direction, the original direction will not change

Figure 8. (a) The clockwise rotation motion; (b) the counterclockwise rotation motion. z

Update local best and global best step: In this step, we update the local best and the global best. If the fitness value of a particle is higher than that of the local best, then the local best will be replaced with the particle; and if the local best is better than the current global best, than we replace the global best with the local best in the swarm. Table 1. The residue fold direction with local search. Direction

Figure 7. All the residues move in the opposite direction.

b) Rotation motion: The structure can rotate clockwise (CW) or counterclockwise (CCW) relative to the local structure, as shown in Figure 8. Figures 8 (a) and 8 (b) represent clockwise rotation and counterclockwise rotation, respectively. The illustrations show that left to right transformation from the top specifically involves the fixed x-axis, the fixed y-axis, and the fixed z-axis. Therefore, we can build Table 1, which lists the relationship between the original directions and the transformed directions. We choose

Right (R) Left (L) Forward (F) Backward (B) Up (U) Down (D) Right (R) Left (L) z

Opposite L R B F D U L R

The z-axis fixed CW CCW B F F B R L L R B F F B U U D D

The y-axis fixed CW CCW D U U D F F B B D U U D R L L R

The x-axis fixed CW CCW R R L L U D D U R R L L B F F B

Update local best and global best step: In this step, we update the local best and the global best. If the fitness value of a particle is higher than that of the local best, then the local best will be replaced with the particle; and if the local best is better than the current global best, than we replace the global best with the

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local best in the swarm. z

Termination Condition: The algorithm is run for a maximum of 50,000 iterations or until minimum free energy is achieved in the sequence. The best member of the population is then returned.

the genetic-based particle swarm optimization mutation process, was able to obtain a new minimal energy value for protein instances 6 and 7. In HGA-PSO, the standard deviation was less than 1. Other algorithms are very sensitive.

4. Simulation Results In this section, our algorithm is compared with the standard genetic algorithm, backtracking-EA [27], aging-AIS [28], and ClonalgI [29]. In Table 2, the 7 chosen HP instances are standard benchmarks used to test the searching ability of the algorithms. The free energy is the optimal or best-known energy value. Hi ,Pi e(…)i indicates i repetitions of the relative symbol or subsequence. Sequences 1 to 7 were introduced in [12]. These sequences have been used as the benchmark for the HP model. Table 2. 3D lattice HP benchmarks. Seq. 1 2 3 4 5 6 7

Length 20 24 25 36 48 50 60

Protein Sequence (HP)2PH(HP)2(PH)2HP(PH)2 H2P2(HP2)6H2 P2HP2(H2P4)3H2 P(P2H2)2P5H5(H2P2)2P2H(HP2)2 P2H(P2H2)2P5H10P6(H2P2)2HP2H5 H2(PH)3PH4PH(P3H)2P4(HP3)2HPH4(PH)3PH2 P(PH3)2H5P3H10PHP3H12P4H6PH2PHP

Energy

-11 -13 -9 -18 -29 -26 -49

We give the structure obtained by our algorithm as follows. Fifty independent runs of the algorithms were performed. Sequences 1 to 4 and sequence 6 used a 100 population size. Sequences 5 and 7 used a 300 population size. The crossover rate and mutation rate were set to 0.8 and 0.3, respectively. For all sequences, 50,000 iterations of our algorithm were run. We also chose c1 = c2 = 1 in Equation (2). These sets of parameters were experimentally determined. The structure of 7 protein sequences can be clearly seen in Fig. 9. In Backtracking-EA [27], the experiments were done with an elitist generational EA (population size = 100, crossover rate = 0.9, mutation rate = 0.01) using linear ranking selection (η=2.0). A maximum number of 105 evaluations were enforced. The Aging-AIS used the standard parameter values k = 10, dup = 2, and c = 0.4, as described in [28]. B cells had the aging parameter τB = 5, with the memory B cells τBm = 10, and a maximum number of evaluations equal to 105. ClonalgI used the 10 individuals in the population. The duplication rate was equal to 4, the mutation rate was equal to 0.6, and the termination criterion was 105 evaluations. The simulation results are summarized in Table 4 in terms of the best found free energy (best), mean, and standard deviation (SD). The HGA-PSO, which adopts

Figure 9. Results of the structure of 7 protein sequences. Table 3. The simulation results obtained from the proposed algorithms compared with the methods given in the literature. Seq. Length

E*

HGA -PSO

GA

1 2 3 4 5 6 7

-11 -13 -9 -18 -29 -26 -49

-11 -13 -9 -18 -29 -26 -49

-11 -13 -9 -18 -25 -23 -37

20 24 25 36 48 50 60

BacktrackAging-EA ing-AIS [27] [28] -11 -13 -9 -18 -25 -23 -39

-11 -13 -9 -18 -29 -23 -41

ClonalgI [29] -11 -13 -9 -18 -29 -26 -48

The simulation results are summarized in Table 4 in terms of the best found free energy (best), mean, and standard deviation (SD). The HGA-PSO, which adopts

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the genetic-based particle swarm optimization mutation process, was able to obtain a new minimal energy value for protein instances 6 and 7. In HGA-PSO, the standard deviation was less than 1. Other algorithms are very sensitive.

[3]

Table 4. Performance of HTGA on the benchmark sequences. No. Size

1 20

Best Mean SD

-11 -11 0

Best Mean SD

-11 -11 0

Best Mean SD

-11 -10.31 0

Best Mean SD

-11 -11 0

Best Mean SD

-11 -10.40 0.57

2 24

3 4 5 25 36 48 HGA-PSO -13 -9 -18 -29 -13 -9 -17.72 -28.88 0 0 0.98 0.47 GA -13 -9 -18 -25 -13 -9 -16.16 -24.22 0 0 1.99 0.60 Backtracking-EA [27] -13 -9 -18 -25 -10.90 -7.98 -14.38 -20.80 0.36 0 0.88 1.17 Aging-AIS [28] -13 -9 -18 -29 13 -9 -16.76 -25.16 0 0 1.02 0.45 ClonalgI [29] -13 -9 -18 -29 -11.26 -8.06 -15.04 -24.20 0.90 0.87 1.37 2.22

6 50

7 60

-26 -25.92 0.27

-49 -48.62 0.59

-23 -22.58 0.66

-37 -36.6 0.48

-23 -20.20 1.15

-39 -34.18 2.00

-23 -22.60 0.40

-41 -39.28 0.24

-26 -23.08 2.05

-48 -42.65 2.74

5. Conclusions

[4]

[5]

[6]

[7]

[8]

This paper proposed an HGA-PSO for solving the protein structure prediction problem. We presented an improved genetic algorithm with mutation based on particle swarm optimization. The HGA-PSO with mutation was based on particle swarm optimization, in which the cognitive component encourages the particles to move toward their own best positions. For long sequences, we found the optimal protein structure with minimum energy. We demonstrate that our algorithm can be applied successfully to the protein folding problem based on the three-dimensional hydrophobic-polar lattice model. Simulation results indicate that our approach performs better than those of existing evolutionary algorithms.

[10]

Acknowledgement

[12]

[9]

[11]

This research is supported by the National Science Council of R.O.C. under grant NSC 99-2221-E-167-022.

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Cheng-Jian Lin received the B.S. degree in electrical engineering from Ta-Tung University, Taiwan, R.O.C., in 1986 and the M.S. and Ph.D. degrees in electrical and control engineering from the National Chiao-Tung University, Taiwan, R.O.C., in 1991 and 1996. Currently, he is a full Professor of Computer Science and Information Engineering Department, National Chin-Yi University of Technology, Taichung County, Taiwan, R.O.C. His current research interests are soft computing, pattern recognition, intelligent control, image processing, bioinformatics, and FPGA design. Shih-Chieh Su received his B.Sc. Degree in Information Management from Ling-Tung University of Technology, Taiwan, R.O.C., in 2006 and his M.Sc. Degree in Computer Science and Information Engineering from Chaoyang University of Technology, Taiwan, R.O.C., in 2006. Currently, he is studying for a PhD in Computer Science and Information in the Engineering Department, National Chung Cheng University, Chiayi County, Taiwan, R.O.C. His current research interests are evolutionary computation, metaheuristic algorithms, and bioinformatics.