# Zigzag Stacks and m-Regular Linear Stacks

@article{Chen2014ZigzagSA, title={Zigzag Stacks and m-Regular Linear Stacks}, author={William Y. C. Chen and Qiang-Hui Guo and Lisa Hui Sun and Jian Wang}, journal={Journal of computational biology : a journal of computational molecular cell biology}, year={2014}, volume={21 12}, pages={ 915-35 } }

The contact map of a protein fold is a graph that represents the patterns of contacts in the fold. It is known that the contact map can be decomposed into stacks and queues. RNA secondary structures are special stacks in which the degree of each vertex is at most one and each arc has length of at least two. Waterman and Smith derived a formula for the number of RNA secondary structures of length n with exactly k arcs. Höner zu Siederdissen et al. developed a folding algorithm for extended RNA…

## 8 Citations

### Enumeration of Extended m-Regular Linear Stacks

- MathematicsJ. Comput. Biol.
- 2016

The study of extended [Formula: see text]-regular linear stacks, in which the degree of each terminal point is bounded by 3, is led to, which is closed to real protein contact maps.

### Combinatorics of Contacts in Protein Contact Maps

- MathematicsBulletin of mathematical biology
- 2018

The generating function for m-regular linear stacks is modified by introducing a new variable y regarding to the number of arcs and an equation satisfied by the generating function is derived of the overallNumber of arcs in m- regular linear stacks with n vertices and k arcs.

### Combinatorics of Contacts in Protein Contact Maps

- MathematicsBulletin of Mathematical Biology
- 2017

Contacts play a fundamental role in the study of protein structure and folding problems. The contact map of a protein can be represented by arranging its amino acids on a horizontal line and drawing…

### Regular Simple Queues of Protein Contact Maps

- MathematicsBulletin of mathematical biology
- 2017

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, are concerned, and a recurrence relation for the generating function of Motzkin paths with $$k_i$$ki peaks at level i is derived.

### On the Number of Saturated and Optimal Extended 2-Regular Simple Stacks in the Nussinov-Jacobson Energy Model

- MathematicsJ. Comput. Biol.
- 2022

It is known that both RNA secondary structure and protein contact map can be presented using combinatorial diagrams, the combinatorial enumeration and related problems of which have been studied…

### Regular Simple Queues of Protein Contact Maps

- MathematicsBulletin of Mathematical Biology
- 2016

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…

### A semi-bijective algorithm for saturated extended 2-regular simple stacks

- Mathematics
- 2021

Combinatorics of biopolymer structures, especially enumeration of various RNA secondary structures and protein contact maps, is of signiﬁcant interest for com-munities of both combinatorics and…

### C O ] 2 4 D ec 2 02 1 A bijective method for saturated extended 2-regular simple stacks in the Nussinov-Jacobson energy model

- Mathematics
- 2021

This paper constructs a bijection between the saturated extended 2 -regular simple stacks and the forests of small trees, which are rooted trees with height one and also called meadows in graph theory.

## References

SHOWING 1-10 OF 28 REFERENCES

### Combinatorics of RNA Secondary Structures with Base Triples

- MathematicsJ. Comput. Biol.
- 2015

The results are of special theoretical interest, because a closer look at the numbers involved suggests that extended RNA secondary structures constitute a new combinatorial class not bijective with any other combinatorsial objects studied so far.

### Combinatorial Properties of RNA Secondary Structures

- MathematicsJ. Comput. Biol.
- 2003

The approach proves to be general enough to compute the average order of a secondary structure together with all the r-th moments and to enumerate substructures such as hairpins or bulges in dependence on the order of the secondary structures considered.

### Contact patterns between helices and strands of sheet define protein folding patterns

- BiologyProteins
- 2007

The essence of protein folding patterns is captured in a concise tableau representation based on the order and contact patterns of secondary structures: helices and strands of sheet that provides a database, derived from the Protein Data Bank, mineable in studies of protein architecture.

### 101 optimal PDB structure alignments: a branch-and-cut algorithm for the maximum contact map overlap problem

- Computer ScienceRECOMB
- 2001

This paper provides the first rigorous algorithm for structure comparison based on developing an effective integer linear programming formulation of protein structure contact maps overlap (CMO), and a branch-and-cut strategy that employs lower-bounding heuristics at the branch nodes.

### Combinatorial Algorithms for Protein Folding in Lattice Models: A Survey of Mathematical Results

- Computer ScienceCommun. Inf. Syst.
- 2009

It is shown how work on 2D self-avoiding walks contact-map decomposition can build upon the exact RNA contacts counting formula by Mike Waterman and collaborators which lead to renewed hope for analytical closed-form approximations for statistical mechanics of protein folding in lattice models.

### STATISTICAL PROPERTIES OF CONTACT MAPS

- Mathematics
- 1999

The number of contact maps corresponding to the possible configurations of a polypeptide chain of N amino acids, represented by (N-1)-step self avoiding walks on a lattice, grows exponentially with N for all dimensions D>1.

### Principles of protein folding — A perspective from simple exact models

- BiologyProtein science : a publication of the Protein Society
- 1995

These studies suggest the possibility of creating “foldable” chain molecules other than proteins, and can account for the properties that characterize protein folding: two‐state cooperativity, secondary and tertiary structures, and multistage folding kinetics.

### Combinatorics of RNA Structures with Pseudoknots

- MathematicsBulletin of mathematical biology
- 2008

The generating function of RNA structures with pseudoknots is derived using a novel 4-term recursion formula and a 2-termRecursion formula for 3-noncrossing RNA structures and RNA secondary structures.

### Lattices for ab initio protein structure prediction

- Computer ScienceProteins
- 2008

A systematic screening of 7 known classic and 11 newly proposed lattices in terms of predictive power was performed and a scale of fitness for all the analyzed lattices was defined, demonstrating that an increase in the coordination number and in the degrees of freedom is necessary but not sufficient to reach the best result.