# Protein folding prediction in the HP model using ions motion optimization with a greedy algorithm

## Abstract

### Background

The function of a protein is determined by its native protein structure. Among many protein prediction methods, the Hydrophobic-Polar (HP) model, an ab initio method, simplifies the protein folding prediction process in order to reduce the prediction complexity.

### Results

In this study, the ions motion optimization (IMO) algorithm was combined with the greedy algorithm (namely IMOG) and implemented to the HP model for the protein folding prediction based on the 2D-triangular-lattice model. Prediction results showed that the integration method IMOG provided a better prediction efficiency in a HP model. Compared to others, our proposed method turned out as superior in its prediction ability and resilience for most of the test sequences. The efficiency of the proposed method was verified by the prediction results. The global search capability and the ability to escape from the local best solution of IMO combined with a local search (greedy algorithm) to the new algorithm IMOG greatly improve the search for the best solution with reliable protein folding prediction.

### Conclusion

Overall, the HP model integrated with IMO and a greedy algorithm as IMOG provides an improved way of protein structure prediction of high stability, high efficiency, and outstanding performance.

## Keywords

Protein folding Ion motion optimization IMOG Hydrophobic-polar (HP) model Global search Local search## Background

Polypeptides consist of a maximum of 20 amino acids. The function of a given protein is determined by the native structure or its polymer structure, which correlates with particular protein functions [1]. The native three-dimensional structure of a protein primarily depends on its amino acid sequence [2]. The development of a highly efficient method for protein folding prediction is in high demand, particularly for protein studies in biotechnology. Currently, several methods have been proposed for protein structure prediction. Comparative modeling and fold recognition approaches commonly use a known protein structure database to train a model in order to classify an unknown protein structure [2]. In contrast, the ab initio method provides a direct prediction using the primary structure or amino acid sequence of a given protein.

Based on the ab initio method, Dill has proposed the hydrophobic-polar protein folding model (HP model) in 1985 which simulates protein folding based on amino acid sequences under the lattice model [3]. In 2013, Bechini used a triangular lattice model for protein folding prediction by simplifying amino acids into hydrophobic (H) and polar (P) types [4]. The predicted space in between the simulated folds is limited to the lattice model; the actual fold space is discrete and the folding of the amino acid sequence follows a self-avoiding walk along the lattice. However, for the protein structure prediction, it remains a challenge to explore the possibility of an extremely large folding in order to obtain an optimal solution for the nondeterministic polynomial-time-hard (NP-hard) problem [5].

Anfinsen’s dogma, following a thermodynamic hypothesis, assumes that the native structure of globular proteins usually folds according to a unique, stable, and kinetically accessible minimum of free energy [6]. The central structure of a globular protein usually contains hydrophobic (non-polar) amino acid compositions that produce hydrophobic attraction to avoid water molecules at the outside. This postulate was also applied to the HP model-based protein structure prediction. To provide an example, if two hydrophobic amino acids are closed together, a hydrophobic-hydrophobic (H-H) interaction is generated. Once the strength of H-H interactions is increased, a more stable structure is predicted. The HP model uses this property giving a negative value for the adjacent hydrophobic amino acids interaction and calculating the number of adjacent hydrophobic interactions [7]. When more adjacent hydrophobic amino acids are present, the predicted structure is closer to the real structure representing optimal protein folding. However, the development of an optimal algorithm for protein folding prediction remains a challenge.

A 2D-triangle-lattice-model [8] is commonly used for 2D HP model of protein folding problem. It has six neighbors in the two dimension triangular lattice on each lattice. When a triangle-lattice-model is embedded in the protein, it can be in topological contact with each other. The vector obtained from the triangle lattice that is easier to model a protein structure on the 2D-triangle-lattice-model [9]. The self-avoiding walk for protein folding is the NP-hard problem and led to several heuristic and meta-heuristic algorithms that were proposed to find best protein structure predictions. These include among others: genetic algorithm [10], branch and bound [11], replica exchange Monte Carlo [12], Evolutionary Monte Carlo [13], greedy-like-algorithm [14].

We use here an ions motion optimization (IMO) algorithm [15] as a heuristic algorithm that is combined – as a novum – with a greedy algorithm for local search within the 2D-triangle-lattice-model to optimize protein folding predictions of high stability, high efficiency, and outstanding performance.

## Methods

We develop here a novel algorithm (IMOG) which combines ions motion optimization (IMO) algorithm [15] with a greedy algorithm as a local search strategy for predicting protein folding reliably at high resolution. Details of our approach are described below.

### Protein folding problem of a 2D-HP-model

*N*amino acids residue sequences of a protein chain, S = {

*a*

_{1},

*a*

_{2}, …,

*a*

_{N}} ⊂ {H, P}. The energy

*E*of a chain conformation was proposed for comparing the stability of several predicted protein structures [7]. The energy

*E*of a chain conformation was defined to the number of H-H contacts

*h*, i.e.,

*E*= − |ε|

*h*, where |ε| is a positive constant. The units of energy

*E*is |ε|. For simplification, the energy

*E*of protein conformation is calculated by the following formula as described previously [9, 10]:

The 2D-HP protein folding problem can be formally defined as finding a conformation of S with minimum energy. This has been proven to be a NP-hard problem.

### Imo

The IMO algorithm has been recently introduced as a metaheuristic optimization technique and it is inspired by the motion of ions [15]. Each single ion in its particular position provides a candidate for solving a particular optimization problem. The movements of ions depend on the attraction or repulsion of ions, i.e., anion (negative charged ion) and cation (positive charged ion). The attraction and repulsion forces between anions and cations are utilized to move the position of ions around a feasible search region. The forces are calculated as acceleration of ion motions. Anions move towards a best fitness of cations and cations move towards a best fitness of anions. Two strategies of ion motion for providing diversification and intensification, these are movements in a liquid phase and a solid phase scenario, respectively. In this study, we implemented an IMO as a heuristic algorithm to find the best HP model for protein folding simulation.

In the 2D-dimensional search space, assuming a population consists of *N* anions/cations moving around. The *i*th anion and *i*th cation are represented by *A*_{i} = (*a*_{i1}, *a*_{i2}, …, *a*_{iD}) and *C*_{i} = (*c*_{i1}, *c*_{i2}, …, *c*_{iD}) in their respective position. The populations of anions and cations are initialized by a uniform random position * A* ∈ {

*A*

_{1},

*A*

_{2}, ..., A

_{N}} and position

*∈ {*

**C***C*

_{1},

*C*

_{2}, ...,

*C*

_{N}}. Each position of an ion provides a candidate solution for a particular problem. When the fitness of evaluation results is calculated, the global best solutions (

*Abest*and

*Cbest*) and current individual worst solutions (

*A*

_{worst}and

*C*

_{worst}) are determined.

In the liquid phase strategy, the attraction forces are used for a search space [15], and its computation is calculated from the distance between two ions (e.g., anion and cation), the measurement is defined as follows:

*AD*

_{i, j}= |

*A*

_{i, j}−

*Cbest*

_{j}| is the distance between the current anion position and the globally best cation position;

*CD*

_{i, j}= |

*C*

_{i, j}−

*Abest*

_{j}| is the distance between the current cation position and the globally best anion position.

*AF*

_{i,j}and

*CF*

_{i,j}represent resultant attraction forces of anions and cations, respectively.

where Φ_{1} and Φ_{2} are random numbers in the range of −1 to 1. *rand*_{1}(), *rand*_{2}() and *rand*_{3}() are random numbers in the range of 0 to 1. *AworstFit* and *CworstFit* are the worst fitness solutions of anion and cation fits, respectively.

### Greedy algorithm

The greedy algorithm is a simple and straightforward heuristic algorithm that makes a current local optimal decision at each stage for global optimization [16]. It is easy to implement and works efficiently depending on the problems although it may or may not be the best approach for solving this task. In any case, it plays a useful role as a optimization method according to its characteristics. Greedy algorithms are widely applied in bioinformatics tools such as among others DNA sequence alignment [16], co-phylogeny reconstruction problem [17], detection of transient calcium signaling [18], resolving the structure and dynamics of biological networks [19].

### IMOG for 2D-HP-model

We implemented the IMO algorithm with a greedy algorithm as a local search strategy for the 2D-HP-model protein folding problem as follows (including IMOG procedure, encoding scheme, fitness function, and improved solid phase strategy):

#### IMOG procedure

- Step 1)
Initialize populations of ions (anions and cations) with random position, each position of an ion is a candidate for the protein folding.

- Step 2)
Estimate the fitness of each ion using energy of the 2D-HP-model according to the Eq. (9).

- Step 3)
Update the global best solution

*Abest*and*Cbest*according to the fitness calculation results. - Step 4)
- Step 5)
If the solid phase condition was satisfied, the solid phase strategies were executed.

- Step 6)
Repeat steps 2–5 until the stop criterion has been met. Consequently, the best protein folding was obtained.

#### Encoding scheme

*A*

_{i}= (

*d*

_{i1},

*d*

_{i2}, …,

*d*

_{iD}) and

*C*

_{i}= (

*d*

_{i1},

*d*

_{i2}, …,

*d*

_{iD}), respectively, where

*d*∈ {1, 2, …, 6} represents six neighbors in the 2D-triangular-lattice-model (Fig. 2a). For example, the IMO encoding the best solution for the sequence HHPPHPHPHPHPHP presents as 4, 5, 6, 2, 6, 2, 1, 3, 2, 4, 3, 5, and 4 (Fig. 2b).

#### Fitness function

*x*

_{i}and

*x*

_{j}of the lattice positions

*p*

_{i}and

*p*

_{j}are respectively indicated to have a H-H contact that is represented as

*contact*(

*p*

_{i},

*p*

_{j}) = − 1, otherwise

*contact*(

*p*

_{i},

*p*

_{j}) = 0. The energy of the protein conformation is defined as the sum of its H-H contacts. If

*S*has an HP sequence, and

*P*=

*p*

_{1},

*p*

_{2}, ...,

*p*

_{n}represents a valid conformation for

*S*, the energy

*E*(

*P*) of

*P*is defined as follows:

*p*

_{4}is not adjacent to

*p*

_{8},

*p*

_{9},

*p*

_{10}and

*p*

_{11}in the amino acid chain but is adjacent in the 2D-model space, thus the fitness is calculated as − 4. Consequently, (

*p*

_{3},

*p*

_{5}), (

*p*

_{3},

*p*

_{11}) and (

*p*

_{5},

*p*

_{8}) are estimated as − 3, non-repetitively. Taken together, the fitness as the energy is − 7.

#### Improved solid phase strategy

*CbestFit*≥

*CworstFit*/ 2 AND

*AbestFit*≥

*AworstFit*/ 2) is unsuitable for our problem. Therefore, we improve the solid phase statement for the HP-model protein folding prediction. In order to enhance the optimal solution, we add the local search method to this phase and pseudo-codes are provided in the formula (10).

where *CbestFitNum* and *AbestFitNum* are the numbers for the global best solution of cation/anion not yet changed. *SolidNum* is a parameter setting for how many times the *CbestFitNum* and *AbestFitNum* were not yet changed.

## Results

### Data sets

_{i}represent the

*i*-fold duplicate of a subsequence. For example, (010)

_{2}represents the original information of data being 010010 (e.g., HPHHPH). Recently, several approaches such as hybrid of hill climbing and genetic algorithm [9], elite-based reproduction strategy-genetic algorithm (ERS-GA) [9], cached divide and conquer evolutionary algorithm [20], co-evolution of memetic algorithms [21], hybrid genetic algorithm (HGA) [22], and tabu search (TS) [23] have been proposed to predict protein folding using benchmark data sets, through HP-model-based computation estimates for protein structure prediction.

First benchmark of amino acids sequences in HP model [9]

Sequence | length | | Amino acids sequence |
---|---|---|---|

1 | 20 | -15 | (101001) |

2 | 24 | −17 | 1(100) |

3 | 25 | −12 | (001) |

4 | 36 | −24 | 0(0011) |

5 | 48 | −43 | 001(0011) |

6 | 50 | −41 | 1(10) |

7 | 60 | – | 001110(1) |

8 | 64 | – | (1) |

Second benchmark of amino acids sequences in HP model [21]

Sequence | length | | Amino acids sequence |
---|---|---|---|

1 | 12 | −11 | 1(10) |

2 | 14 | −11 | 1100(10) |

3 | 14 | −11 | 1(100) |

4 | 16 | −11 | 110(100) |

5 | 16 | −11 | 1(100) |

6 | 17 | −11 | 1(100) |

7 | 17 | −17 | 1(11) |

8 | 20 | − 17 | 1(100) |

9 | 20 | −17 | 1(10) |

10 | 21 | −17 | 1(100) |

11 | 21 | −17 | 110(100) |

12 | 21 | −17 | 1100(10) |

13 | 22 | −17 | 1(100) |

14 | 23 | −25 | 11(10) |

15 | 24 | −17 | 1(100) |

16 | 24 | −25 | 11(10) |

17 | 24 | −25 | 11(10) |

18 | 30 | −25 | 11(100) |

19 | 30 | −25 | 11(100) |

20 | 37 | −29 | 11(100) |

### Parameter settings

The advantages of the IMOG algorithm is a fewer number of tuning parameters, only population size (e.g., number of anion and cation) and iteration size. Here, we set 100 for the numbers of anions and cations and 2000 for the iteration size in this study. Additionally, the mutation probability of a greedy algorithm for each point in the local search is 0.25.

### Comparison of the best prediction

Comparison of algorithms studied here for optimal solutions

Sequence | SGA | HGA | TS | ERS-GA | HHGA | IMOG |
---|---|---|---|---|---|---|

1 | −11 | | | | | |

2 | −10 | −13 | | − 13 | | |

3 | − 10 | − 10 | | −12 | | |

4 | − 16 | −19 | | − 20 | − 23 | |

5 | −26 | −32 | − 40 | − 32 | | −40 |

6 | −21 | −23 | NA | −30 | −38 | |

7 | −40 | − 46 | | − 55 | − 66 | − 67 |

8 | − 33 | −46 | −50 | − 47 | − 63 | |

Comparison of the best prediction results of IMO with MMA algorithm

Sequence* | MMA | IMOG | Sequence* | MMA | IMOG |
---|---|---|---|---|---|

1 | NA | | 11 | | |

2 | | | 12 | | |

3 | | | 13 | | |

4 | | | 14 | | |

5 | | | 15 | −16 | |

6 | | | 16 | | |

7 | | | 17 | | |

8 | | | 18 | −24 | |

9 | | | 19 | −24 | |

10 | | | 20 | −26 | |

### Comparsion of stability

Comparison of the best solutions and stabilities with other algorithms

Sequence* | | ERS-GA | HHGA | IMOG | |||
---|---|---|---|---|---|---|---|

Best | Mean | Best | Mean | Best | Mean | ||

1 | −15 | | −12.50 | | −14.73 | | − 14.73 |

2 | − 17 | − 13 | −10.20 | | −14.93 | | −14.93 |

3 | − 12 | | − 8.47 | | − 11.57 | | − 11.57 |

4 | − 24 | −20 | − 16.17 | −23 | −21.27 | − 23 | −21.27 |

5 | −43 | −32 | −28.13 | | − 37.30 | | − 37.30 |

6 | −41 | − 30 | − 25.30 | −38 | −34.10 | − 38 | −34.10 |

7 | – | −55 | −49.43 | − 66 | − 61.83 | − 66 | −61.83 |

8 | – | −47 | −42.37 | −63 | − 56.53 | −63 | −56.53 |

## Discussion

The IMO algorithm [15] is a population-based algorithm designed according to the natural properties of ions. Its idea is to divide the ion population into negative and positive charged ions (i.e., anions and cations). It is based on the fact that anions repel anions but attract cations and cations repel cations but attracts anions. It is reported that IMO is very competitive in solving challenging optimization problems [15]. Moreover, the greedy algorithm is also reported to improve local searches [25]. In computer science, hybrid algorithms are commonly applied in solving optimization problems [26, 27, 28, 29, 30, 31, 32]. Accordingly, we developed a novel algorithm that combines the IMO algorithm [15] with a greedy algorithm we here name IMOG for protein folding prediction. The key concept of our proposed IMOG algorithm is based on the characteristics of IMO having global search capabilities while escaping from the local best solution. In addition, the greedy algorithm is used in each update to strengthen its local search ability.

In this paper, two phases (liquid and solid) were designed for diverse and intense search that can make sure convergence of the ions toward an optimum in the feasible space and resolve local optima trap. Our proposed method has redundant extra parameters and it adapts itself automatically to search spaces. The obtained results indicate that the integrated algorithm has a good search ability and stability. Compared with other methods, the stability and search ability of our proposed method is better than other methods for protein structure prediction for most of the test sequences.

The HP-model of protein structure prediction problem was developed as discrete problem in folding space. In HP model, the amino acids were classified into hydrophobic and polar that keeps the prediction complexity down. Nevertheless, the whole possible combinations of protein folding prediction problem is still complex. Recently, researchers assume that the simple optimization algorithms were hard to solve protein folding structure prediction effectively [21]. Accordingly, many improved algorithms were proposed to enhance ability of prediction in HP model problem, such as HHGA [9]. The HHGA is an effective algorithm which combines genetic algorithm with a hill climbing algorithm, it can solve longer amino acid sequence well performance.

In this study, we implement an IMO with a greedy algorithm as local search for a 2D-HP model protein folding problem. The technical behavior (liquid phase strategy) of IMO is similar to the particle swarm optimization (PSO) [33] algorithm but the IMO had improved the “particle” to divide into two parts as anion and cation. The two global superior solutions were utilized to search global optimal solutions. It also had a mechanism to escape local optima through the solid phase strategy. We improved the IMO in order to enhance seeking local optima by adding a greedy algorithm to the solid phase strategy. Consequently, our proposed IMOG algorithm has several advantages including low computational complexity, rapid convergence, a smaller number of tuning parameters, avoidance of local optima and superior performance in searching for global optima [15].

Recently, several protein structure prediction systems were developed. For example, Rosetta [34, 35, 36] and i-TASSER [37, 38] are sophisticated comprehensive software suites for protein structure and function prediction. Structure prediction with Rosetta was reported to be enhanced performance with an additional modeling, such as the combined covalent-electrostatic model of hydrogen bonding [34]. The processing that generates protein structure and function predictions by i-TASSER is firstly retrieved from protein data bank (PDB) library by Local Meta-Threading-Server (LOMETS) [39]. When LOMETS is unable to identify suitable template, i-TASSER will process the ab initio modeling for protein structure and identify the low free energy states by SPICKER [40]. It is possible that our proposed IMO may support the function of SPICKER and i-TASSER by the calculation of energy mentioned in the current study. It warrants further evaluating the performance that our proposed IMO algorithm combines with Rosetta and i-TASSER for protein folding prediction in the future.

There are some limitations in the current study. The longest length of test sequence is 64 amino acids and it has 6^{64} possible combinations in 2D triangular lattice model with six neighbors, showing superior to other test algorithms [9, 21, 22, 23]. However, the performance of our proposed IMO algorithm is only based on 28 test data sets. It warrants further evaluating for more data sets and longer length of test protein sequences. It is noted that our proposed IMO method is based on the relative energy. For precise comparison, the absolute free energy for protein folding structure warrants further investigation in the future.

## Conclusions

This study uses an ab initio technique (hydrophobic polar model) to predict protein structures. This is one of the most commonly applied methods for protein structure prediction. We propose and develop here a combination of the IMO with a greedy algorithm for protein folding predictions assuming a hydrophobic polar model. Experimental results show that our proposed IMOG method can reliably seek and find the best solution among short sequences, and also effectively obtain satisfying results with longer sequences. Taken together, these results demonstrate that the hybrid algorithm, combining the IMO algorithm with a greedy algorithm provides a useful tool for protein folding predictions.

## Notes

### Acknowledgements

The authors thank our colleague Dr. Hans-Uwe Dahms for English editing.

### Funding

This work was supported by funds of the Ministry of Science and Technology, Taiwan (MOST 102–2221-E-151-024-MY3, MOST 105-2221-E-992-307-MY2, MOST 106-2221-E-992-327-MY2, and MOST 107–2320-B-037-016), the National Sun Yat-sen University-KMU Joint Research Project (#NSYSU-KMU 107-p001), and the Health and welfare surcharge of tobacco products, the Ministry of Health and Welfare, Taiwan, Republic of China (MOHW107-TDU-B-212-114016).

### Authors’ contributions

L-YC and H-WC conceived and designed the research and wrote the draft manuscript. C-HY instructed K-CW and Y-SL for algorithm processing. C-HY and H-WC revised the manuscript. All authors read and approved the final manuscript.

### Ethics approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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