# GRSA Enhanced for Protein Folding Problem in the Case of Peptides

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Background

- To design the physical code that aims to determine the interatomic forces of the protein structure for a given amino acid sequence.
- To solve the computational problem of designing an algorithm to predict the native structure from a given amino acid sequence.
- To perform an algorithm for the folding process by nature, which rapidly finds the routes or pathways from an initial solution to the NS or functional structure.

#### 2.1. Computational Methods in PFP

#### 2.2. Simulated Annealing

Algorithm 1 Classical Simulated Annealing. |

1: SA (${T}_{i}$, ${T}_{fp}$, ${T}_{f}$, $\alpha $) |

2: ${T}_{k}={T}_{i}$ |

3: ${S}_{old}=generateSolution()$ |

4: while ${T}_{k}\ge {T}_{f}$ do |

5: while $Metropolis$ do |

6: ${S}_{new}=perturbation\left({S}_{old}\right)$ |

7: $\Delta E=E\left({S}_{new}\right)-E\left({S}_{old}\right)$ |

8: if $\Delta E\le 0$ then |

9: ${S}_{new}={S}_{old}$ |

10: else if ${e}^{-\Delta E/{T}_{i}}>random\left[0;1\right)$ then |

11: ${S}_{old}={S}_{new}$ |

12: end |

13: end |

14: ${T}_{k+1}=\alpha \ast {T}_{k}$ |

15: end |

16: end |

#### 2.3. Chemical Reaction Optimization

- Unimolecular collisions: When the molecule hits the wall of the container.
- Intermolecular collision: When a molecule collides with other molecules.

- Unimolecular collision (low energy collisions). In this group, we find two reactions:
- On-wall ineffective collision is established as follows [39]:“It represents the situation when a molecule collides with a wall of the container and then bounces away remaining in one single unit”.In GRSA2, the current solution ${S}_{old}$ is changed by a new solution ${S}_{new}$ obtained by a perturbation function. This operation is equivalent to the classical SA perturbation. Thus, the complexity of GRSA2 is not modified. This operation is implemented in line seven of Algorithm 3, which calls the function soft perturbation or Algorithm 2, explained in Section 4. As we will see, this operation does not add complexity to the classical SA.
- Decomposition. In this case, a molecule (solution) hits a wall and then is divided into several parts. In the GRSA2 algorithm, decomposition is a perturbation operation that generates two new solutions from the current solution. This perturbation is implemented in GRSA2 in lines sixth and seven of Algorithms 2 and 3, respectively. Again, to include this operation in SA for obtaining GRSA2 does not increase the complexity of the new algorithm.

- Intermolecular collision (high energy collisions). This collision has the next elementary reactions:
- Intermolecular ineffective collision. These kinds of reactions occur when multiple molecules collide with each other and then bounce away. The number of molecules remains the same.
- Synthesis. In this reaction, several molecules are fused into a single one.

#### 2.4. Analytical Tuning Method

## 3. Ab Initio Definition

#### 3.1. Ab Initio Problem in PFP

- Given a sequence of $n$ amino acids; ${a}_{1},{a}_{2},{a}_{3},\dots ,{a}_{n}$, which represents the primary structure of a protein with a set of dihedral angles $\sigma =\left\{{\sigma}_{1},{\sigma}_{2},{\sigma}_{3},\dots ,{\sigma}_{m}\right\}$, and an energy function $f\left({\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{m}\right)$ which represents the free energy or Gibbs energy (G).
- Find the native structure of the protein, such that $f\left(\sigma \right)$ represents the minimum energy value, where the optimal solution $\sigma $ defines the best three-dimensional configuration. The PFP variables are the set $\sigma $ of dihedral angles.

- Phi $\left(\varphi \right)$ is the angle between the amino group and the alpha carbon.
- Psi $\left(\psi \right)$ is the angle between the alpha carbon and the carboxyl group.
- Omega $\left(\omega \right)$ is defined for each two consecutive amino acids.
- Chi $\left(\chi \right)$ is defined between the two planes conformed by two consecutive carbon atoms in the radical group.

#### 3.2. Force Field

- ${r}_{ij}$ is the distance in $\u212b$ between the atoms $i$ and $j$.
- ${A}_{ij},{B}_{ij},{C}_{ij}$ and ${D}_{ij}$ are the parameters of the empirical potentials.
- ${q}_{i}$ and ${q}_{j}$ are the partial charges on the atoms $i$ and $j$, respectively.
- $\epsilon $ is the dielectric constant, which is usually set to $\epsilon =2$.
- 332 is a factor used to obtain the energy in kcal/mol.
- ${U}_{n}$ is the energetic torsion barrier of rotation about the bond $n$.
- ${k}_{n}$ is the multiplicity of the torsion angle ${\phi}_{n}$.

## 4. An Enhancement of Golden Ratio Simulated Annealing

#### 4.1. The Enhancement of GRSA

_{0}constant determined by the final and initial temperature ${T}_{f}$ and ${T}_{0}$ respectively. Thus, for any specific instance, the number of iterations is a function of the $\alpha $ parameter; then, the execution time decreases by reducing this parameter. To give an idea, with $\alpha =0.7$, the time required will be less than 15% of the time used for SA with a normal cooling scheme (using $\alpha =0.95$). On Figure 2 the exact proportion is represented by ${n}_{SA-0.70}/{n}_{SA-0.95}$, and is given as 14.38%. Notice that this relation considers the complete iterations from the ${T}_{0}$ to ${T}_{f}$ temperatures. In other words, ${n}_{SA-0.7}/{n}_{SA-0.95}$ gives the proportion of randomly executed SA two times, both of them until SA converges.

- GRSA with one cut-off temperature:
- The processing time of phase A is $-{C}_{0}/Ln0.7$ multiplied by the fraction of iterations where this phase is executed $\left(1-0.618\right)$. Let ${\mu}_{1}$ be the proportion of time of phase A concerning the normal execution time of SA ($\alpha =0.95$); as is shown in Figure 2, ${\mu}_{1}=5.48\%$ of ${n}_{SA-0.95}$.
- The processing time of phase B is given by $\left[-C/Ln0.95\right]\times 0.618$. Now, the time proportion of phase B for the normal execution of SA is ${\mu}_{2}=61.8\%$ of ${n}_{SA-0.95}$.
- The total proportion of GRSA processing time compared to SA is ${\mu}_{1}+{\mu}_{2}=67.28\%$.

- GRSA with two or more cut-off temperatures
- Phase A is the same process as case 1 (with $\alpha =0.7$) and uses ${\mu}_{1}=5.48\%$ of ${n}_{SA-0.95}$.
- Phase B is divided into nGolden sections. For instance, if nGolden equals 2, phase B is divided into two subphases. The new $\alpha $ values for the next subsections can be again 0.7 and 0.95 for the next subphases. In other words, each time a subdivision is made, the last subphase will have a new $\alpha $ parameter equal to 0.95. The division process continues until nGolden parameter is reached. When nGolden equals 2, the two new subphases (with $\alpha =0.70$ and $\alpha =0.95$) will have ${\mu}_{1}+{\mu}_{2}=67.28\%$ of the execution time of phase B. The proportion of the total processing time (time of phase A plus time of new subphases generated from B) will be ${\left(0.6728\right)}^{2}+0.0548=50.7\%$ of the execution time of SA.
- When nGolden is increased, a reduction of the time is obtained.
- The alpha values can be changed in several ways. Instead of using the last numbers (0.7, 0.95) to divide the subsections, a linear or exponential function for the alphas can be used. In our case, the linear approach was used [14], which gives similar reductions to those previously presented. Experimentation reveals that, in general, a nGolden value lower or equal to five gives good results in the case of peptides.

#### 4.2. Soft Perturbation in GRSA

Algorithm 2 Soft perturbation. |

1: SoftPertubation (${S}_{old}$) |

2: $moleColl,b$ |

3: if $b>moleColl$ then |

4: Randomly select one particle $M\omega $ |

5: if $Decompositioncriterionmet$ then |

6: $Decomposition()$ |

7: else if |

8: $SoftCollision()$ |

9: end |

10: end |

11: end |

_{f}) is established and the slope (m) of a set of energies is calculated, and when m is lower than a tolerance parameter ε (0.001 in our case), the algorithm is ended. Finally, the alpha value (α) is updated depending on how many golden sections (defined by the variable nGolden) are used. Note that the number of cuts of temperature (T

_{fp}) is indirectly defined in Algorithm 3 by the nGolden parameter are used.

Algorithm 3 GRSA2. |

1: GRSA2 (${T}_{i}$, ${T}_{fp}$, ${T}_{f}$, E, S, $\alpha $, KE, EP nGolden) |

2: ${T}_{k}={T}_{i}$ |

3: ${S}_{old}=generateSolution()$ |

4: while ${T}_{k}\ge {T}_{f}$ do |

5: ${T}_{fp}={T}_{fp}*\Phi $ |

6: while $Metropolis$ do |

7: ${S}_{new}=SoftPertubation\left({S}_{old}\right)$ |

8: EP = Enew |

9: if $EP\le Eold+KE$ then |

10: ${S}_{old}={S}_{new}$ |

11: KE = ((Eold+KE)-EP)*random [0, 1] |

12: end if |

13: end while |

14: if ${T}_{k}\le {T}_{f}$ then |

15: $m=slope()$ |

16: if $m<\epsilon $ then |

17: ${T}_{k}={T}_{f}$ (the algorithm is stopped) |

18: end if |

19: end if |

20: if $\left({T}_{k}\le {T}_{fp}\mathrm{or}\text{}\mathrm{nGolden}\right)$ then |

21: $\alpha ={\alpha}_{new}$ |

22: ${T}_{k+1}=\alpha *{T}_{k}$ |

23: else |

24: ${T}_{k+1}=\alpha *{T}_{k}$ |

25: end if |

26: end while |

27: end |

## 5. Results

#### 5.1. Experimental Description

#### 5.2. Results and Discussion

_{1}(n), f

_{2}(n), and f

_{3}(n) of the three algorithms, SA, GRSA, and GRSA2, respectively. Because f

_{2}(n) and f

_{3}(n) are lower or equal than f

_{1}(n), they belong to the same complexity class [60].

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Number | Instance (PDB Code) | Number of Amino Acids | Number of Variables |
---|---|---|---|

1 | 2LWC | 5 | 19 |

2 | 1EGS | 9 | 49 |

3 | 1UAO | 10 | 47 |

4 | 1L3Q | 12 | 62 |

5 | 2EVQ | 12 | 66 |

6 | 1IN3 | 12 | 74 |

7 | 1RNU | 13 | 68 |

8 | 1LCX | 13 | 81 |

9 | 1GJF | 14 | 79 |

10 | 1K43 | 14 | 84 |

11 | 2BTA | 15 | 100 |

12 | 1LE3 | 16 | 91 |

13 | 1PEF | 18 | 124 |

14 | 1L2Y | 20 | 100 |

15 | 1DU1 | 210 | 134 |

16 | 1PEI | 22 | 143 |

17 | 1WZ4 | 23 | 123 |

18 | 2MLT | 26 | 158 |

19 | 1T0C | 31 | 132 |

Instances | Average Energy (kcal/mol) | Average RMSD | Average TM-Score |
---|---|---|---|

2LWC | −7.7386 | 0.5538 | 0.5007 |

1EGS | −1.0498 | 2.9325 | 0.2816 |

1UAO | −34.2519 | 2.7139 | 0.2818 |

1L3Q | −49.5822 | 4.2446 | 0.2116 |

2EVQ | −53.3023 | 1.5843 | 0.2663 |

1IN3 | −70.1176 | 3.6054 | 0.2748 |

1RNU | −73.6159 | 1.4122 | 0.3526 |

1LCX | −61.9788 | 1.3277 | 0.2436 |

1GJF | −67.6448 | 1.76 | 0.2820 |

1K43 | −74.1248 | 2.46 | 0.2276 |

2BTA | −98.6907 | 3.3561 | 0.1992 |

1LE3 | −78.0697 | 2.0468 | 0.1791 |

1PEF | −68.1363 | 1.9766 | 0.1780 |

1L2Y | −92.8494 | 2.126 | 0.1805 |

1DU1 | −123.4410 | 2.0280 | 0.1760 |

1PEI | −111.8189 | 2.351 | 0.1435 |

1WZ4 | −112.8309 | 2.75 | 0.1572 |

2MLT | −86.3540 | 2.8553 | 0.1666 |

1T0C | −109.1762 | 3.1829 | 0.1970 |

Instances | Average Energy (kcal/mol) | Average RMSD | Average TM-Score |
---|---|---|---|

2LWC | −5.7567 | 0.5593 | 0.4970 |

1EGS | 3.5779 | 2.2703 | 0.2830 |

1UAO | −49.4173 | 1.1766 | 0.2718 |

1L3Q | −66.6739 | 2.784 | 0.2203 |

2EVQ | −69.7577 | 1.5208 | 0.2576 |

1IN3 | −96.1027 | 1.2333 | 0.3469 |

1RNU | −70.9097 | 1.4382 | 0.2534 |

1LCX | −60.4809 | 1.5791 | 0.2205 |

1GJF | −93.3798 | 1.2517 | 0.3989 |

1K43 | −98.7355 | 1.9287 | 0.1730 |

2BTA | −153.3692 | 2.6587 | 0.2075 |

1LE3 | −93.4192 | 1.89333 | 0.1773 |

1PEF | −57.2994 | 2.0026 | 0.1534 |

1L2Y | −125.3933 | 2.4276 | 0.1734 |

1DU1 | −134.8380 | 1.5084 | 0.1695 |

1PEI | −114.1452 | 2.315 | 0.1936 |

1WZ4 | −125.0288 | 2.0323 | 0.1453 |

2MLT | −150.0441 | 2.1519 | 0.2899 |

1T0C | −110.1145 | 3.5264 | 0.1999 |

Instances | RMSD GRSA2 | RMSD PEP-FOLD3 | TM-Score GRSA2 | TM-Score PEP-FOLD3 |
---|---|---|---|---|

2LWC | 0.134 | 0.49915802 | 0.622022 | 0.63645887 |

1EGS | 0.174 | 0.73379194 | 0.363588 | 0.28297143 |

1UAO | 0.218 | 1.43239212 | 0.379374 | 0.40506025 |

1L3Q | 0.49 | 2.11590502 | 0.304162 | 0.24278709 |

2EVQ | 0.842 | 0.82452263 | 0.332682 | 0.46217599 |

1IN3 | 0.604 | 0.92708461 | 0.436492 | 0.39695857 |

1RNU | 0.352 | 0.80774343 | 0.435094 | 0.62276608 |

1LCX | 0.552 | 1.22937939 | 0.287596 | 0.33622833 |

1GJF | 0.308 | 0.65046896 | 0.562328 | 0.58219463 |

1K43 | 0.782 | 1.50581118 | 0.258046 | 0.33411994 |

2BTA | 0.594 | 2.43201208 | 0.27246 | 0.18155674 |

1LE3 | 0.826 | 1.96238744 | 0.263946 | 0.24700389 |

1PEF | 0.712 | 0.61298789 | 0.20271 | 0.66990523 |

1L2Y | 1.312 | 1.86484044 | 0.247734 | 0.3428772 |

1DU1 | 1.286 | 1.29916825 | 0.256142 | 0.25837997 |

1PEI | 1.198 | 1.29391279 | 0.313088 | 0.35394815 |

1WZ4 | 3.034 | 2.74149027 | 0.191944 | 0.23998161 |

2MLT | 0.972 | 1.57230256 | 0.462832 | 0.43948739 |

1T0C | 0.352 | 3.21218634 | 0.435094 | 0.22636347 |

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**MDPI and ACS Style**

Frausto-Solís, J.; Sánchez-Hernández, J.P.; Maldonado-Nava, F.G.; González-Barbosa, J.J.
GRSA Enhanced for Protein Folding Problem in the Case of Peptides. *Axioms* **2019**, *8*, 136.
https://doi.org/10.3390/axioms8040136

**AMA Style**

Frausto-Solís J, Sánchez-Hernández JP, Maldonado-Nava FG, González-Barbosa JJ.
GRSA Enhanced for Protein Folding Problem in the Case of Peptides. *Axioms*. 2019; 8(4):136.
https://doi.org/10.3390/axioms8040136

**Chicago/Turabian Style**

Frausto-Solís, Juan, Juan Paulo Sánchez-Hernández, Fanny G. Maldonado-Nava, and Juan J. González-Barbosa.
2019. "GRSA Enhanced for Protein Folding Problem in the Case of Peptides" *Axioms* 8, no. 4: 136.
https://doi.org/10.3390/axioms8040136