Markov Chain-Based Sampling for Exploring RNA Secondary Structure under the Nearest Neighbor Thermodynamic Model and Extended Applications
Ribonucleic acid (RNA) secondary structures and branching properties are important for determining functional ramifications in biology. While energy minimization of the Nearest Neighbor Thermodynamic Model (NNTM) is commonly used to identify such properties (number of hairpins, maximum ladder distan...
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doaj-79a62d9d9315400a871f5aaa06d7c3472020-11-25T03:46:43ZengMDPI AGMathematical and Computational Applications1300-686X2297-87472020-10-0125676710.3390/mca25040067Markov Chain-Based Sampling for Exploring RNA Secondary Structure under the Nearest Neighbor Thermodynamic Model and Extended ApplicationsAnna Kirkpatrick0Kalen Patton1Prasad Tetali2Cassie Mitchell3School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USASchool of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USASchool of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USADepartment of Biomedical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USARibonucleic acid (RNA) secondary structures and branching properties are important for determining functional ramifications in biology. While energy minimization of the Nearest Neighbor Thermodynamic Model (NNTM) is commonly used to identify such properties (number of hairpins, maximum ladder distance, etc.), it is difficult to know whether the resultant values fall within expected dispersion thresholds for a given energy function. The goal of this study was to construct a Markov chain capable of examining the dispersion of RNA secondary structures and branching properties obtained from NNTM energy function minimization independent of a specific nucleotide sequence. Plane trees are studied as a model for RNA secondary structure, with energy assigned to each tree based on the NNTM, and a corresponding Gibbs distribution is defined on the trees. Through a bijection between plane trees and 2-Motzkin paths, a Markov chain converging to the Gibbs distribution is constructed, and fast mixing time is established by estimating the spectral gap of the chain. The spectral gap estimate is obtained through a series of decompositions of the chain and also by building on known mixing time results for other chains on Dyck paths. The resulting algorithm can be used as a tool for exploring the branching structure of RNA, especially for long sequences, and to examine branching structure dependence on energy model parameters. Full exposition is provided for the mathematical techniques used with the expectation that these techniques will prove useful in bioinformatics, computational biology, and additional extended applications.https://www.mdpi.com/2297-8747/25/4/67Markov chain Monte CarloRNA secondary structurenearest neighbor thermodynamic ModelMarkov chain convergence |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Anna Kirkpatrick Kalen Patton Prasad Tetali Cassie Mitchell |
spellingShingle |
Anna Kirkpatrick Kalen Patton Prasad Tetali Cassie Mitchell Markov Chain-Based Sampling for Exploring RNA Secondary Structure under the Nearest Neighbor Thermodynamic Model and Extended Applications Mathematical and Computational Applications Markov chain Monte Carlo RNA secondary structure nearest neighbor thermodynamic Model Markov chain convergence |
author_facet |
Anna Kirkpatrick Kalen Patton Prasad Tetali Cassie Mitchell |
author_sort |
Anna Kirkpatrick |
title |
Markov Chain-Based Sampling for Exploring RNA Secondary Structure under the Nearest Neighbor Thermodynamic Model and Extended Applications |
title_short |
Markov Chain-Based Sampling for Exploring RNA Secondary Structure under the Nearest Neighbor Thermodynamic Model and Extended Applications |
title_full |
Markov Chain-Based Sampling for Exploring RNA Secondary Structure under the Nearest Neighbor Thermodynamic Model and Extended Applications |
title_fullStr |
Markov Chain-Based Sampling for Exploring RNA Secondary Structure under the Nearest Neighbor Thermodynamic Model and Extended Applications |
title_full_unstemmed |
Markov Chain-Based Sampling for Exploring RNA Secondary Structure under the Nearest Neighbor Thermodynamic Model and Extended Applications |
title_sort |
markov chain-based sampling for exploring rna secondary structure under the nearest neighbor thermodynamic model and extended applications |
publisher |
MDPI AG |
series |
Mathematical and Computational Applications |
issn |
1300-686X 2297-8747 |
publishDate |
2020-10-01 |
description |
Ribonucleic acid (RNA) secondary structures and branching properties are important for determining functional ramifications in biology. While energy minimization of the Nearest Neighbor Thermodynamic Model (NNTM) is commonly used to identify such properties (number of hairpins, maximum ladder distance, etc.), it is difficult to know whether the resultant values fall within expected dispersion thresholds for a given energy function. The goal of this study was to construct a Markov chain capable of examining the dispersion of RNA secondary structures and branching properties obtained from NNTM energy function minimization independent of a specific nucleotide sequence. Plane trees are studied as a model for RNA secondary structure, with energy assigned to each tree based on the NNTM, and a corresponding Gibbs distribution is defined on the trees. Through a bijection between plane trees and 2-Motzkin paths, a Markov chain converging to the Gibbs distribution is constructed, and fast mixing time is established by estimating the spectral gap of the chain. The spectral gap estimate is obtained through a series of decompositions of the chain and also by building on known mixing time results for other chains on Dyck paths. The resulting algorithm can be used as a tool for exploring the branching structure of RNA, especially for long sequences, and to examine branching structure dependence on energy model parameters. Full exposition is provided for the mathematical techniques used with the expectation that these techniques will prove useful in bioinformatics, computational biology, and additional extended applications. |
topic |
Markov chain Monte Carlo RNA secondary structure nearest neighbor thermodynamic Model Markov chain convergence |
url |
https://www.mdpi.com/2297-8747/25/4/67 |
work_keys_str_mv |
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