Regular Simple Queues of Protein Contact Maps

A protein fold can be viewed as a self-avoiding walk in certain lattice model, and its contact map is a graph that represents the patterns of contacts in the fold. Goldman, Istrail, and Papadimitriou showed that a contact map in the 2D square lattice can be decomposed into at most two stacks and one...

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Published in:	Bulletin of mathematical biology Vol. 79; no. 1; pp. 21 - 35
Main Authors:	Guo, Qiang-Hui, Sun, Lisa Hui, Wang, Jian
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.01.2017 Springer Nature B.V
Subjects:	Algorithms Cell Biology Life Sciences Mathematical and Computational Biology Mathematical Concepts Mathematics Mathematics and Statistics Models, Molecular Nucleic Acid Conformation Original Article Protein Folding Protein Interaction Mapping - statistics & numerical data Protein Interaction Maps Proteins - chemistry RNA - chemistry Motzkin path 92B05 Contact map Queue 05A15 Stack
ISSN:	0092-8240, 1522-9602, 1522-9602
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Abstract	A protein fold can be viewed as a self-avoiding walk in certain lattice model, and its contact map is a graph that represents the patterns of contacts in the fold. Goldman, Istrail, and Papadimitriou showed that a contact map in the 2D square lattice can be decomposed into at most two stacks and one queue. In the terminology of combinatorics, stacks and queues are noncrossing and nonnesting partitions, respectively. In this paper, we are concerned with 2-regular and 3-regular simple queues, for which the degree of each vertex is at most one and the arc lengths are at least 2 and 3, respectively. We show that 2-regular simple queues are in one-to-one correspondence with hill-free Motzkin paths, which have been enumerated by Barcucci, Pergola, Pinzani, and Rinaldi by using the Enumerating Combinatorial Objects method. We derive a recurrence relation for the generating function of Motzkin paths with k i peaks at level i , which reduces to the generating function for hill-free Motzkin paths. Moreover, we show that 3-regular simple queues are in one-to-one correspondence with Motzkin paths avoiding certain patterns. Then we obtain a formula for the generating function of 3-regular simple queues. Asymptotic formulas for 2-regular and 3-regular simple queues are derived based on the generating functions.
AbstractList	(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) A protein fold can be viewed as a self-avoiding walk in certain lattice model, and its contact map is a graph that represents the patterns of contacts in the fold. Goldman, Istrail, and Papadimitriou showed that a contact map in the 2D square lattice can be decomposed into at most two stacks and one queue. In the terminology of combinatorics, stacks and queues are noncrossing and nonnesting partitions, respectively. In this paper, we are concerned with 2-regular and 3-regular simple queues, for which the degree of each vertex is at most one and the arc lengths are at least 2 and 3, respectively. We show that 2-regular simple queues are in one-to-one correspondence with hill-free Motzkin paths, which have been enumerated by Barcucci, Pergola, Pinzani, and Rinaldi by using the Enumerating Combinatorial Objects method. We derive a recurrence relation for the generating function of Motzkin paths with ... peaks at level i, which reduces to the generating function for hill-free Motzkin paths. Moreover, we show that 3-regular simple queues are in one-to-one correspondence with Motzkin paths avoiding certain patterns. Then we obtain a formula for the generating function of 3-regular simple queues. Asymptotic formulas for 2-regular and 3-regular simple queues are derived based on the generating functions. A protein fold can be viewed as a self-avoiding walk in certain lattice model, and its contact map is a graph that represents the patterns of contacts in the fold. Goldman, Istrail, and Papadimitriou showed that a contact map in the 2D square lattice can be decomposed into at most two stacks and one queue. In the terminology of combinatorics, stacks and queues are noncrossing and nonnesting partitions, respectively. In this paper, we are concerned with 2-regular and 3-regular simple queues, for which the degree of each vertex is at most one and the arc lengths are at least 2 and 3, respectively. We show that 2-regular simple queues are in one-to-one correspondence with hill-free Motzkin paths, which have been enumerated by Barcucci, Pergola, Pinzani, and Rinaldi by using the Enumerating Combinatorial Objects method. We derive a recurrence relation for the generating function of Motzkin paths with k i peaks at level i , which reduces to the generating function for hill-free Motzkin paths. Moreover, we show that 3-regular simple queues are in one-to-one correspondence with Motzkin paths avoiding certain patterns. Then we obtain a formula for the generating function of 3-regular simple queues. Asymptotic formulas for 2-regular and 3-regular simple queues are derived based on the generating functions. A protein fold can be viewed as a self-avoiding walk in certain lattice model, and its contact map is a graph that represents the patterns of contacts in the fold. Goldman, Istrail, and Papadimitriou showed that a contact map in the 2D square lattice can be decomposed into at most two stacks and one queue. In the terminology of combinatorics, stacks and queues are noncrossing and nonnesting partitions, respectively. In this paper, we are concerned with 2-regular and 3-regular simple queues, for which the degree of each vertex is at most one and the arc lengths are at least 2 and 3, respectively. We show that 2-regular simple queues are in one-to-one correspondence with hill-free Motzkin paths, which have been enumerated by Barcucci, Pergola, Pinzani, and Rinaldi by using the Enumerating Combinatorial Objects method. We derive a recurrence relation for the generating function of Motzkin paths with [Formula: see text] peaks at level i, which reduces to the generating function for hill-free Motzkin paths. Moreover, we show that 3-regular simple queues are in one-to-one correspondence with Motzkin paths avoiding certain patterns. Then we obtain a formula for the generating function of 3-regular simple queues. Asymptotic formulas for 2-regular and 3-regular simple queues are derived based on the generating functions. A protein fold can be viewed as a self-avoiding walk in certain lattice model, and its contact map is a graph that represents the patterns of contacts in the fold. Goldman, Istrail, and Papadimitriou showed that a contact map in the 2D square lattice can be decomposed into at most two stacks and one queue. In the terminology of combinatorics, stacks and queues are noncrossing and nonnesting partitions, respectively. In this paper, we are concerned with 2-regular and 3-regular simple queues, for which the degree of each vertex is at most one and the arc lengths are at least 2 and 3, respectively. We show that 2-regular simple queues are in one-to-one correspondence with hill-free Motzkin paths, which have been enumerated by Barcucci, Pergola, Pinzani, and Rinaldi by using the Enumerating Combinatorial Objects method. We derive a recurrence relation for the generating function of Motzkin paths with [Formula: see text] peaks at level i, which reduces to the generating function for hill-free Motzkin paths. Moreover, we show that 3-regular simple queues are in one-to-one correspondence with Motzkin paths avoiding certain patterns. Then we obtain a formula for the generating function of 3-regular simple queues. Asymptotic formulas for 2-regular and 3-regular simple queues are derived based on the generating functions.A protein fold can be viewed as a self-avoiding walk in certain lattice model, and its contact map is a graph that represents the patterns of contacts in the fold. Goldman, Istrail, and Papadimitriou showed that a contact map in the 2D square lattice can be decomposed into at most two stacks and one queue. In the terminology of combinatorics, stacks and queues are noncrossing and nonnesting partitions, respectively. In this paper, we are concerned with 2-regular and 3-regular simple queues, for which the degree of each vertex is at most one and the arc lengths are at least 2 and 3, respectively. We show that 2-regular simple queues are in one-to-one correspondence with hill-free Motzkin paths, which have been enumerated by Barcucci, Pergola, Pinzani, and Rinaldi by using the Enumerating Combinatorial Objects method. We derive a recurrence relation for the generating function of Motzkin paths with [Formula: see text] peaks at level i, which reduces to the generating function for hill-free Motzkin paths. Moreover, we show that 3-regular simple queues are in one-to-one correspondence with Motzkin paths avoiding certain patterns. Then we obtain a formula for the generating function of 3-regular simple queues. Asymptotic formulas for 2-regular and 3-regular simple queues are derived based on the generating functions.
Author	Guo, Qiang-Hui Wang, Jian Sun, Lisa Hui
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CitedBy_id	crossref_primary_10_1089_cmb_2021_0421 crossref_primary_10_1016_j_aam_2024_102722 crossref_primary_10_1016_j_aam_2023_102491 crossref_primary_10_1016_j_disc_2024_114317
Cites_doi	10.1089/cmb.2014.0133 10.1089/cmb.2007.0004 10.1016/0097-3165(77)90020-6 10.1016/j.tcs.2003.10.037 10.1017/CBO9780511609589 10.1016/S0378-4371(00)00410-6 10.1016/S1359-0278(97)00041-2 10.1007/s00285-012-0594-x 10.1016/S0166-218X(97)00118-2 10.1093/bioinformatics/btr220 10.1137/0403019 10.1016/j.ejc.2011.09.039 10.1016/j.disc.2004.04.001 10.1016/0166-218X(92)00038-N 10.1016/j.ejc.2004.02.009 10.1090/S0002-9947-06-04210-3 10.1007/s11538-007-9240-y 10.1007/s11538-007-9265-2 10.1089/cmb.2013.0022 10.1145/369133.369199 10.4310/CIS.2009.v9.n4.a2 10.1109/SFFCS.1999.814624
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Snippet	A protein fold can be viewed as a self-avoiding walk in certain lattice model, and its contact map is a graph that represents the patterns of contacts in the... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) A protein fold can be viewed as a self-avoiding walk in certain lattice model, and... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).A protein fold can be viewed as a self-avoiding walk in certain lattice model, and...
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SubjectTerms	Algorithms Cell Biology Life Sciences Mathematical and Computational Biology Mathematical Concepts Mathematics Mathematics and Statistics Models, Molecular Nucleic Acid Conformation Original Article Protein Folding Protein Interaction Mapping - statistics & numerical data Protein Interaction Maps Proteins - chemistry RNA - chemistry
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Title	Regular Simple Queues of Protein Contact Maps
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