# A Novel Efficient Simulated Annealing Algorithm for the RNA Secondary Structure Predicting with Pseudoknots

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## Abstract

The pseudoknot structure of RNA molecular plays an important role in cell function. However, existing algorithms cannot predict pseudoknots structure efficiently. In this paper, we propose a novel simulated annealing algorithm to predict nucleic acid secondary structure with pseudoknots. Firstly, all possible maximum successive complementary base pairs would be identified and maintained. Secondary, the new neighboring state could be generated by choosing one of these successive base pairs randomly. Thirdly, the annealing schedule is selected to systematically decrease the temperature as the algorithm proceeds, the final solution is the structure with minimum free energy. Furthermore, the performance of our algorithm is evaluated by the instances from PseudoBase database, and compared with state-of-the-art algorithms. The comparison results show that our algorithm is more accurate and competitive with higher sensitivity and specificity indicators.

## Keywords

RNA secondary structure Pseudoknot Simulated annealing algorithm Minimum free energy## 1 Introduction

Pseudoknots usually contains not well-nested base pairs, as shown in Fig. 1(b). These non-nested base pairs make the presence of pseudoknots in RNA sequences more difficult to predict by dynamic programming, which use a recursive scoring system to identify paired stems. The general problem of predicting minimum free energy structures with pseudoknots has been shown to be NP-complete [3].

The dynamic programming (DP) is the first computational approach used to predict RNA structure [4, 5, 6, 7, 8]. It can be seen that the temporal and spatial complexity of the prediction algorithm for dynamic programming is high, which is not good for the algorithm to make predictions for long sequence because it will take more time and resources. The other prediction approaches are based on heuristic methods and thermodynamics models [9, 10, 11, 12, 13].

In this paper, we propose a novel efficient simulated annealing algorithm to predict nucleic acid secondary structure with pseudoknots. The performance of our algorithm compared with RNA structure method using PseudoBase [14] benchmark instances. The comparison result shows that our algorithm is more accurate and competitive with higher sensitivity and specificity values.

## 2 Problem Defines

For a given RNA sequence *X* = 5’−*x*_{1}*x*_{2}…*x*_{n}−3’ of length n, *M*(*X*) is the mapping string of complementary base-pairs of *X*, *M*(*X*) = (*m*_{1}, *m*_{2}, …, *m*_{i}, …, *m*_{n}). Each *m*_{i} corresponds to the form of \( \left( {i,j,k} \right) \), which is called *k* successive base pairs, where \( i \) and \( j \) are the base position, where *k* is the number of successive base pair, and two constraints must be satisfied:

### Base Pairs Constraint:

If \( \left( {i,j,k} \right) \in M \), then {(*x*_{i}*, x*_{j}), (*x*_{i+1}*, x*_{j-1}), …, (*x*_{i+k-1}*, x*_{j-k+1})} \( \in \) {(A, U), (G, C), (G, U)} in RNA.

### K Successive Base Pairs Constraint:

*MinStem*is the minimum number of stack and

*MinLoop*is the minimum number of loop (Fig. 2). Such as there must be at least \( Minloop \) unpaired bases in a hairpin loop.

## 3 The Proposed Approach

### 3.1 Set of K Successive Base Pairs

*MinStem*and

*MinLoop*parameters. Assume that there are three variables i, j, k, which \( i \) and \( j \) are the base position, where k is the number of successive base pair. According to the above Fig. 2, we can be seen that i, j, k need to satisfy the following three constraints:

### 3.2 Evaluation Function

For most MFE based RNA secondary structure prediction algorithm, the complex thermodynamic model is often used to evaluate candidate solutions [15]. There are no useful information to guide the candidate solution to find lower neighbor energy state. Consequently, the convergence of these MFE based prediction algorithms is very slow. However, among all of the secondary structure, only the successive base pairs stack structure \( \Delta G_{S} \) provide negative free energy which contributes to the reduction of free energy. The stability of RNA sequence can also be approximately evaluated by successive base pairs stacks.

*M*(

*X*) is the mapping string of complementary base pairs of

*X*,

*M*(

*X*) = (

*m*

_{1},

*m*

_{2}, …,

*m*

_{i}, …,

*m*

_{n}). Each

*m*

_{i}corresponds to the form of \( \left( {i,j,k} \right) \), where

*m*

_{i}.

*k*equals

*k*,

*group*is the number of stems, then the following formula:

*PseudoknotGroup*is the predicted number of pseudoknot by the algorithm, and

*MaxPesudoKnot*is the expected number of pseudoknot.

### 3.3 Overall Algorithm

## 4 Experiments Result

*sensitivity*(SN) and

*specificity*(SP) [19], as shown in Eq. (8).

Where TP represents the number of correctly predicted base pairs; FP represents the number of incorrectly predicted base pairs; FN represents the number of unpredicted base pairs compared to the known structure. When the prediction results are accurate, both SN and SP should be close to 100*%*.

*%*and 84.3

*%*respectively.

Comparison results with sensitivity and specificity indicator

Sequences | [18] | IPknot | TT2NE | OPA | [18] | IPknot | TT2NE | OPA |
---|---|---|---|---|---|---|---|---|

Sensitivity | Specificity | |||||||

Ec_PK3 | 85.7 | 71.4 | | 92.9 | | 76.9 | | 92.9 |

BEV | 93.8 | 81.3 | 87.5 | | | 81.3 | 66.7 | 76.2 |

BaEV | 86.7 | 0.0 | | 93.3 | | 0.0 | 65.2 | 70.0 |

VMV | 100.0 | 50.0 | 92.9 | | | 38.9 | 65.0 | 70.0 |

ALFV | 100.0 | 64.7 | | | | 45.8 | 70.8 | 70.8 |

SARS-CoV | 69.2 | 69.2 | 51.7 | | 72.0 | 78.3 | 46.9 | |

BCRV1 | 96.7 | 76.7 | | | 85.3 | 82.1 | | |

AMV3 | 71.8 | 74.4 | 74.4 | | 80.0 | 96.7 | 72.5 | |

RSV | 97.4 | 71.8 | | 92.3 | 88.4 | 90.3 | | 90.0 |

CCMV3 | 66.7 | | 71.1 | 73.3 | 66.7 | | 71.1 | 76.7 |

| 86.8 | 71.5 | 87.5 | | 82.1 | 75.4 | 74.6 | |

## 5 Conclusion

This paper proposes efficient SA algorithm for the RNA secondary structure predicting with pseudoknots, combined with the evaluation function to compensate for the high time complexity of the free energy calculation model. The algorithm sets the *MinStem* and *MinLoop* parameters to determine the pseudoknot structure formed by the base pair cross-combination, and optimizes the pool of candidate solutions, thereby reducing the time cost of the algorithm. We use the evaluation function to further reduce the time consumption of RNA secondary structure prediction algorithms. Moreover, the performance of our algorithm is compared with state of art algorithms using ten PseudoBase benchmark instances, and the comparison result shows that our algorithm is more accurate and competitive with higher sensitivity and specificity values.

## Notes

### Acknowledgement

This work was supported by the National Natural Science Foundation of China (Grant No. 61472293). Research Project of Hubei Provincial Department of Education (Grant No. 2016238).

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