Bioimpacts. 11(4):271-279. doi: 10.34172/bi.2021.37

Original Research

GADP-align: A genetic algorithm and dynamic programming-based method for structural alignment of proteins

Soraya Mirzaei 1ORCID logo, Jafar Razmara 1, *ORCID logo, Shahriar Lotfi 1
1Department of Computer Science, Faculty of Mathematics, Statistics, and Computer Science, University of Tabriz, Tabriz, Iran
*Corresponding author: Jafar Razmara, Email: razmara@tabrizu.ac.ir


Introduction: Similarity analysis of protein structure is considered as a fundamental step to give insight into the relationships between proteins. The primary step in structural alignment is looking for the optimal correspondence between residues of two structures to optimize the scoring function. An exhaustive search for finding such a correspondence between two structures is intractable.

Methods: In this paper, a hybrid method is proposed, namely GADP-align, for pairwise protein structure alignment. The proposed method looks for an optimal alignment using a hybrid method based on a genetic algorithm and an iterative dynamic programming technique. To this end, the method first creates an initial map of correspondence between secondary structure elements (SSEs) of two proteins. Then, a genetic algorithm combined with an iterative dynamic programming algorithm is employed to optimize the alignment.

Results: The GADP-align algorithm was employed to align 10 ‘difficult to align’ protein pairs in order to evaluate its performance. The experimental study shows that the proposed hybrid method produces highly accurate alignments in comparison with the methods using exactly the dynamic programming technique. Furthermore, the proposed method prevents the local optimal traps caused by the unsuitable initial guess of the corresponding residues.

Conclusion: The findings of this paper demonstrate that employing the genetic algorithm along with the dynamic programming technique yields highly accurate alignments between a protein pair by exploring the global alignment and avoiding trapping in local alignments.

Keywords: Bioinformatics, Protein structure alignment, Genetic algorithm, Dynamic programming


© 2021 The Author(s)
This work is published by BioImpacts as an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by-nc/4.0/). Non-commercial uses of the work are permitted, provided the original work is properly cited.


Structural alignment of proteins plays a fundamental role in understanding their functional similarity and evolutionary relationship. Generally, since‌ protein structure is more conserved than protein sequence, structure comparison is more evident to identify high distantly proteins than sequence comparison. As a result, the prediction of the new protein’s function is carried out through the detection of local or global structural similarity between a new protein and a protein with a known function. 1 The methods are used to measure the structural similarity between two proteins and then produce a pairwise structural alignment between them.

Many algorithms have been developed to find the optimal structural alignment. These methods deal with two significant problems: a search algorithm for determining an optimal alignment between two structures and a scoring function to evaluate the created alignment. 2 The structural alignment problem can be formulated as a search algorithm for finding the optimal set of correspondences minimizing the spatial distance between pairs of residues. 3,4 The algorithm does not have any initial knowledge about correspondence between residues. Accordingly, the problem is computationally intractable and known as NP-hard. 5 Though many similarity scoring functions have been proposed in polynomial time. 6,7 no procedure (of any running time) has been proposed to optimize structural alignment. 8,9 Taylor and coworkers write: ‘In structure comparison, we do not even have an algorithm that guarantees an optimal answer for pairs of structures’. 10

Many efforts have been made to explore efficient pairwise structural alignment. In this regard, SSAP 11 employs a double dynamic programming technique based on atom-to-atom vectors in the structure space. SPalign 12 is another method for pairwise protein structure alignment based on SPscore as a size-independent scoring function to compare protein sequences. Alternatively, several methods use the distance geometry to represent each protein by a pairwise distance matrix between all Cα atoms. DALI 13 is a popular method that uses the Monte Carlo algorithm to obtain the optimal superposition of matrices. Other methods use additional heuristics to produce faster or more accurate alignment, such as CE. 14 MMLigne 15 uses statistical inference and CLICK 16 matches the clique of residues to find one-to-one equivalence residues. Several strategies have been proposed to optimize the previously obtained equivalence set further. These include the Monte Carlo algorithm or simulated annealing, 13 dynamic programming, 8 the incremental combinatorial extension of the optimal path, 14,17 and genetic algorithm. 18-20 Recently, the multi-criteria protein structure comparison tool is developed for combining methods and generating consensus structural similarity scores. 21 The tools additionally are used to generate the training set for a template-based protein structure prediction based on the threading strategy. These methods commonly employ machine learning techniques 22,23 for implementation of their proposed strategy. Recently, some approaches have been proposed to solve the protein classification problem based on two-dimensional multi-view images of the protein structure and learning methods. 32,33

Several methods use iterative dynamic programming over an initial map of corresponding residues to find a transformation and choose the final alignment after some heuristic iterations. STRUCTAL 24 uses five sets of equivalent residues, including start, middle, and end of a chain, sequence identity, and similar Ca torsion angles, to produce the final alignment based on the highest score. TM-align 9 obtains an initial alignment by aligning the secondary structures, and gapless or gapped matching of two structures. SPalign 25 uses similar heuristics as TM-align and gapless threading, secondary structure fitting, and fragment with 20 size matching are used as initial alignment. TS-AMIR 26 uses a text modeling-based technique to match secondary structure elements (SSEs) and then find the starting and ending residues of these elements as the initial alignment. These strategies for optimizing the initial correspondence would be impractical in some conditions like when sequence identity is low or when the SSEs have different sizes. In this paper, we introduce GADP-align as a hybrid algorithm for the optimal alignment of proteins by combining genetic algorithm with iterative dynamic programming technique. The genetic algorithm is a heuristic algorithm that is highly adaptable with dynamic programming. In the proposed method, in addition to the leading operators of genetic algorithm, we add the shift operator for enhancing the genetic algorithm’s global exploring along two protein chains. The main objective of combining these two optimization methods is to explore the global alignment and to avoid trapping in local alignment originated by improper initial alignment.

The paper is organized as follows. Section 2 describes the combination of GA with iterative dynamic programming algorithm in detail. Section 3 shows the evaluated genetic algorithm performance and makes a comparison between TM-align, MMLigne, SPalign, and CLICK methods. Section 4 concludes the proposed algorithm.

Materials and Methods

The GADP-align algorithm

The proposed GADP-align algorithm is introduced in this section. represents the steps of the algorithm in summary. The algorithm first looks for a match between SSEs of two proteins to produce an initial match between their secondary structures. Then, the algorithm employs a procedure as shown in a flowchart to align two structures at the residue level.

Fig. 1. The GADP-align algorithm in summary.

Matching of the SSE sequences

To find an initial corresponding element between the secondary structures of two proteins, the algorithm encodes each protein secondary structure in an SSE sequence including two letters of the alphabet (H for α-helix and S for β-strand). Then, it employs the Needleman-Wunsch dynamic programming algorithm 27 to create an initial correspondence between two sequences. The algorithm considers a score of +2 for identical SSEs and -1 for non-identical SSEs as well as a score of -2 for the gap-opening penalty. Then, the initial match is submitted to the genetic algorithm to search heuristically for initial corresponding residues within each matched SSEs.

The genetic algorithm

The algorithm starts with a set of chromosomes as the initial map of SSE matching. represents the structure of a sample chromosome. Each chromosome contains a sequence of matched SSE pairs, and each pair is represented by the SSE type (0 for α-helix and 1 for β-stand), the number of corresponding residues within the SSE, and the list of their position along with the peptide. Coil and loop structures are highly irregular and are ignored in both protein structures. The population size was set to N=100 by default.

Fig. 2. The structure of a sample chromosome for aligning 1BGE:B (HHHHH) and 2GMF:A (HHSHHHSH) proteins. The rectangles with diagonal lines represent α-helices and the rest boxes are β-strands. The first SSE of 2GMF:A protein is matched with the second SSE of 1BGE:B using the Needleman-Wunsch algorithm and seven residues are selected randomly as the initial correspondence of these matched SSEs.

Initial population

To produce an initial population, the SSE sequences of two proteins are compared and matched using Needleman-Wunsch dynamic programming, and a set of chromosomes is defined based on pairs of matched SSEs. For each pair of matched SSEs in the individuals, an initial list of corresponding residues is randomly selected, ranging between 20% and 100% of the shorter SSE length.

Fitness function

For evaluating each alignment, the TM-Score is used via the formula 9 :


Where LTarget is the length of query protein which is the protein with a lower number of SSEs, Lali is the number of aligned residues, di is the distance between ith pair of aligned residues, and. d0(LTarget)=1.24Ltarget1531.8 . It can be seen that the formula is independent of the protein size.

Selection strategy

The selection process chooses the high-fitness chromosome by using the tournament selection. This means that the chromosome with a higher fitness value has a higher probability of being placed in the intermediate population. The tournament selection randomly selects kindividuals through the substitution of the current population. The best chromosome having the highest score is entered into the middle population. Thisprocess is repeated N (=100)times. In the experiments, the value of kwas set to 3.

Crossover operator

The crossover operator is used to combine the genetic information of two individuals. In the proposed method, the crossover is done at two points along the chromosome with Pc=0.75 as the crossover probability. These two points are chosen randomly, and then, the first and third segments are swapped, as shown in .

Fig. 3. The crossover operator. The SSEs alongside diagonal lines are swapped.

Mutation operator

The mutation operator randomly mutates an individual. It allows the algorithm to search within the solution space and converge the population to the global maximum while it prevents the algorithm from falling into a local optimal trap. In the proposed method, the number of aligned residues within a pair of matched SSEs in the individual may be increased or decreased by 1 with Pm=0.04 as the mutation probability.

Shift operator

The procedure utilizes the shift operator to find an optimal matching between the SSE sequences of two proteins. The operator generates a new matching between SSEs and tries to converge to the global optimal matching instead of a local matching. For each individual in the population, the SSEs are shifted left or right with the shifting probability of Ps=0.45. shows the shift operator for the proteins 3HHR and 1TEN PDB-ID, whereas the SSEs of 1TEN is shifted right or left along with the 3HHR protein.

Fig. 4. The shift operator. Dashed boxes represent the matched SSEs (before applying the shift operator). After two shifts to the left, a new SSE matching has been generated as represented in dark gray.


The generational replacement strategy was used to replace the entire population of the current generation with the generated population after applying the crossover, mutation, and shift operations. To prevent missing the individual with the highest score, the best chromosome is passed to the next generation as an elitism.


Since the genetic algorithm is a random search approach, the algorithm does not generate an exact solution. Two different criteria were used for the termination of the algorithm. The first criterion is the unchanged maximum score (elitism) for 30 numbers of generation, and the second criterion is the termination of the algorithm after the production of 100 numbers of generation.

Dynamic programming

In each iteration, the algorithm submits the chromosome to a dynamic programming procedure to compute and apply the Kabsch rotation matrix. 28 As a result, a new alignment is created by running the dynamic programming on the score similarity matrix, which is defined as:

Eq. (2) S(i,j)=11+dij2/d0(Lmin)2

where dij is the distance of the i and j denote residues from the query and target proteins and d0(Lmin)=1.24Lmin1531.8 with Lmin is the length of the smaller protein. Furthermore, the gap opening penalty of dynamic programming is defined -0.6. Then, the new rotation matrix and score matrix are computed based on newly aligned residues. This procedure is repeated for n(=5) times to obtain the alignment with the highest score. Algorithm 1 shows the pseudo-code implemented for the above iterative dynamic programming algorithm.

Algorithm 1. Iterative dynamic programming

Input : a chromosome

Output : the alignment with the highest score

1: AR ←aligned residues in chromosome k

2: for i=1 to n do

3: Compute Kabsch rotation matrix based on AR

4: Rotate the target protein using the rotation matrix

5: Compute the score similarity matrix using formula 2

6: Run dynamic programming on the score similarity matrix

7: AR←new aligned residue pairs

8: end for


The proposed method was implemented using the C++ programming language within visual studio 2013 on a personal computer having 2.60 GHz Core i5 and 6 GB RAM. The genetic algorithm was evaluated forits convergence and stability. The performance of the proposed method was examined on different datasets and compared with similar state-of-the-art methods, including TM-align, MMLigne, SPalign, and CLICK. In this section, the results have been presented and discussed.

Convergence analysis

The implemented genetic algorithm has been investigated for its convergence to an optimal or near-optimal solution. represents the convergence graph drawn for three different protein pairs. In each graph, the highest fitness function values are shown by red curves, while the blue curves represent the average fitness values of the generations. As it can be seen from the figure, the fitness curves are swinging during iterations indicating that different initial correspondence causes different values of the scoring function.

Fig. 5. The convergence of the algorithm to an optimal solution for (A) 1A8Y & 1A81, (B) 2RHE & 3HLA:B, and (C) 2GMF:A & 1BGE:B. The blue swinging curve shows the average fitness of generations, and the red steps curve represents the best fitness value in the generation.

Stability analysis

Regarding that genetic algorithms are stochastic search methods, it is necessary to run the algorithm multiple times to examine its stability. A standard deviation of less than average concludes the stability of the algorithm. shows the stability diagram for three different protein pairs. The TM-score for alignment of 2GMF:A & 1BGE:B is the same for all 20 runs. However, for 1TEN & 3HHR:B and 1A8Y & 1A81 pairs of proteins, the standard deviation differs for different runs. Additionally, for analyzing the stability and effectiveness of randomized algorithms like genetic algorithm, it is essential to use statistical tests such as t test and the Wilcoxon signed-rank test, which are parametric and non-parametric methods, respectively. If the data is approximately normally distributed, the t test would be more reliable to be used. However, the Wilcoxon signed-rank test does not assume the distribution of the data. We use Shapiro-Wilk and Kolmogorov-Smirnov test to assess whether the data is normal or not. Table 1 shows the Shapiro-Wilk and Kolmogorov-Smirnov normality tests obtained by the proposed method. From the results, both tests have a p-value lower than 0.05, which indicates data are not normally distributed. Thus, it is recommended to use a non-parametric statistical test method. The Wilcoxon signed-rank test is used to compare the average of two related samples and assess their difference. A null hypothesis Ho is typically defined to state that there is no median difference between pairs of samples and H1 is considered as an alternative hypothesis whether the median difference is not equal to zero. The P value of the statistical test denotes whether we accept Ho or not. The significant level of α=0.05 is considered for rejecting Ho. The last column in Table 1 shows the results of the Wilcoxon signed-rank test. The asymp. sig. (2-tailed) the value represents the P value of the test. The results indicate that the P value is better than the significance level α (0.05 by default). This observation reveals that there is no statistically significant difference between the first and second 15 runs of the genetic algorithms. As a result, the stability of the genetic algorithm is proven.

Fig. 6. GADP-align stability diagram of (A) 1TEN & 3HHR:B, (B) 1A8Y & 1A81, and (C) 2GMF:A &1BGE:B

Table 1. The results of the normality test of four instances in 30 runs of the genetic algorithm
Query Protein Target Protein Kolmogorov-Smirnov Shapiro-Wilk Wilcoxon signed-rank
Statistic df* Sig. Statistic df Sig. Z Asymp. Sig (2 tailed)

*df: The degree of freedom.

Experimental result

To evaluate the performance of the GADP-align algorithm, 10 ‘difficult to align’ protein pairs were used. 29 The results were compared with those of the TM-align algorithm as a state-of-the-art alignment method. In general, the quality of alignment depends on the contradictory necessities of earning a lower RMSD and a higher length of the alignment. TM-score is a reasonable particular score to evaluate the alignment quality by making a balance between the alignment length and its accuracy. As shown in Table 2, GADP-align yields a higher TM-score than TM-align in four cases among ten protein pairs. TM-align could not align significantly with two protein pairs, including 1TEN & 3HHR:B and 2RHE & 3HLA:B whereas the TM-score is less than the threshold of 0.5. This is while GADP-align produces a more significant alignment in terms of TM-score for these two protein pairs. Besides, the results of GADP-align were compared with those of TM-align, MMLigne, SPalign, and CLICK as four online structure alignment tools. Table 3 shows the results in terms of root mean square deviation (RMSD) and length of alignment (Nali). As shown in the table, the proposed method produces a higher length of alignment in all 10 cases than MMLigne, SPalign, and CLICK. The method also produces alignments better than or equal to those of TM-align for nine protein pairs in terms of length of the alignment.

Table 2. Comparison of structure alignments for 10 ‘difficult’ structures obtained by TM-align and GADP-align methods
Query Protein Target Protein TM-align GADP-align
TM-1 TM-2 TM-1 TM-2
1UBQ1FXI:A0.568410.4793 0.5864 0.49831
1TEN3HHR:B0.26850.16816 0.81947 0.40575
2RHE3HLA:B0.461380.23485 0.53542 0.48751
1PAZ2AZA:A0.559690.52731 0.55984 0.52735
2GMF:A1BGE:B0.591830.48049 0.62034 0.50198

TM-1: TM-Score which is normalized by the length of Chain 1.

TM-2: TM-Score which is normalized by the length of Chain 2.

Table 3. Comparison of structure alignments for 10 ‘difficult’ structures obtained by TM-align, MMLigner, SPalign, and CLICK methods
Query Protein Target Protein TM-align 9 MMLigner 15 SPalign 25 CLICK 16 GADP-align
RMSD N ali RMSD N ali RMSD N ali RMSD N ali RMSD N ali
1UBQ1FXI:A2.63632.80077572.48621.9158 2.63 63
1TEN3HHR:B5.14511.6533684 1.74 86 1.4782 1.74 86
2RHE3HLA:B3.39793.13553792.73772.2266 3.68 86
1PAZ2AZA:A2.82862.12664772.39831.9279 3.00 87
1MOL:A1CEW:I2.25821.91291772.12811.5772 2.25 82
2RHE1CID2.901002.50037902.56971.7181 2.90 100
1EDE1CRL4.322353.057881612.862022.32213 4.32 235
1NSB:A2SIM 3.82 312 2.808252732.862862.252423.79311
2GMF:A1BGE:B3.441033.01005962.70941.8976 3.80 110
4FGF1TIE 2.82 117 2.731931112.751152.0395 2.82 117

Regarding the low differences between the numbers of SSEs of protein pairs in the Fischer set, GADP-align has not remarkably produced different TM-score values except for two cases. Therefore, another experiment was organized using a set of 200 non-homologous protein chains, which were collected by Zhang and Skolnick. 9 Ten pairs of proteins were randomly chosen from the set and GADP-align as well as the above four alignment tools were employed for their alignment. The alignment results are represented in Table 4, including RMSD and the length of the alignment. Besides, the value of TM-Score was calculated for both GADP-align and TM-align methods. The results for GADP-align in Table 4 are the average values calculated based on ten runs of the developed genetic algorithm. As shown in Table 4, the alignment quality in terms of TM-score by GADP-align is higher than that of TM-align for three protein pairs, while both methods obtained a similar TM-score for four cases. In the case of non-homologous protein pair 1ACZ & 1A02N, the TM-Score value obtained by GADP-align (=0.27) is considerably lower than that of TM-align (=0.49 that is very near to similarity threshold of 0.5) indicating that GADP-align precisely identifies non-homologous proteins by using an appropriate initial correspondence map between a protein pair. From Table 4, it can be seen that the alignment quality in terms of RMSD and length of alignment by GADP-align is higher than MMLigne, SPalign, and CLICK methods for the non-homologous protein pairs.

Table 4. Comparison of structure alignments of selecting randomly five proteins of a set of 200 non-homologous protein chains
Query Protein Target Protein MMLigner 15 SPalign 25 CLICK 16 TM-align 9 GADP-align
RMSD N ali RMSD N ali RMSD N ali RMSD N ali TM-score 1 RMSD N ali TM-score 1
TM-score 2 TM-score 2
1A02:N1AOE:A2.05224.26672.251025.75108 0.23
5.54111 0.26
1A811A8Y - - 4.47992.11886.43119 0.25
6.59139 0.29
1A02:N1A81 - - 4.71902.381376.20124 0.26
6.22125 0.26
1ACZ1A02:N3.52793.01762.391223.2979 0.49
5.49114 0.27
1A811ACZ - - 3.51482.291224.5596 0.18
6.54120 0.14
1ACZ1A8Y4.39493.72492.491224.9668 0.32
6.58124 0.21
1ACZ1AOE:A1.73233.69522.431105.4267 0.28
6.23120 0.32
1AOEA1A81 - -3.79712.41845.33105 0.32
5.53106 0.31
1A8Y1A02:N - - 4.12922.341246.06123 0.23
6.52131 0.23
1AOE:A1A8Y3.80563.5166 2.22756.09118 0.32
6.56118 0.32

Nali: number of aligned residues, RMSD: root-mean-square deviation

TM-score 1:TM-Score which is normalized by length of Chain 1

TM-score 2: TM-Score which is normalized by length of Chain 2


According to the previous comparative analysis, 30,31 conventional methods are not powerful enough for protein structure alignment. Despite the proposition of different methods based on effective biological insights, they mostly employ an unsuitable scoring scheme to assess the quality of alignment. The scoring schemes are commonly work based on two contradictory criteria including RMSD and the length of the alignment. TM-score makes a balance between these two scores in order to provide a reasonable measure for assessing the quality of alignment.

The analysis of the experimental results conducted in this study indicates the effectiveness of the GADP-align method for protein structure alignment. The overall results show the method aligns difficult to align pairs of proteins with a quality better than or equal to other state-of-the-art tools. In addition, examining the methods on a set of 200 non-homologous protein chains demonstrates the high applicability of the method in comparison with other similar tools.

GADP-align combines the advantages of the global exploring ability of the genetic algorithms and fast convergence mechanism of dynamic programming. This combination makes the method free from the typical requirement of the user-supplied initial guess to achieve the optimal alignment. In this way, the method automatically generates a set of initial residue-residue equivalences using a genetic algorithm, and then, searches between sets of residue-residue correspondence maximizing the scoring function. Furthermore, the method iteratively looks for alternative SSE matching instead of relying on the SSE matching initially produced by the Needleman-Wunsch algorithm. GADP-align looks for the optimal alignment of residues within the matched SSEs through a procedure of randomly choosing the aligned residues. The results in Table ‌4 depict that the methods, which exclusively use the iterative dynamic programming algorithm with an arbitrary initial alignment, converge to the nearest local minimum RMSD.


The developed genetic algorithm utilizes the novel shift operator, particularly in structures with high differences in size to avoid trapping of the search in a local optimal score. Besides, the results demonstrate that a relevant initial guess of corresponding residues is essential to obtain alignment with a high score. Since the protein structure alignment is a discrete optimization problem, other efficient evolutionary algorithms which are suitable for discrete optimization can be employed instead of genetic algorithms.

Funding sources


Ethical statement

The authors declare no ethical issue to be considered.

Competing interests

The authors declare no conflicts of interest.

Authors’ contribution

SM, JR, SL: conceptualization, writing and reviewing; SM, JR; data handling; SM, JR, SL: experiments design; SM, JR: data analysis; SM, JR, SL: provision of study materials and equipment; SM, JR, SL: study validation; SM: data presentation and draft preparation; JR: supervision and project administration; SL: study consultation.

Research Highlights

What is the current knowledge?
  • Three-dimensional structure of query and target proteins

  • Optimization by dynamic programming approach

  • Optimization by genetic algorithms

What is new here?
  • Combining genetic algorithm and dynamic programming helps to explore the global alignment and avoid trapping in local alignments

  • The shift operator helps to find an optimal matching between the secondary structure elements.


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Submitted: 03 Feb 2020
Revised: 10 Jun 2020
Accepted: 16 Jun 2020
First published online: 08 Jul 2020
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