Zigzag Stacks and m-Regular Linear Stacks

  title={Zigzag Stacks and m-Regular Linear Stacks},
  author={William Y. C. Chen and Qianghui Guo and Lisa Hui Sun and Jian Wang},
  journal={Journal of computational biology : a journal of computational molecular cell biology},
  volume={21 12},
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… 

Enumeration of Extended m-Regular Linear Stacks

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

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.

Regular Simple Queues of Protein Contact Maps

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

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

A bijective method for saturated extended 2-regular simple stacks in the Nussinov-Jacobson energy model

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.



Combinatorics of RNA Secondary Structures with Base Triples

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

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

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

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

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.


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

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.

Mining Protein Contact Maps

This paper focuses on two main tasks: given the database of protein sequences, discover an extensive set of non-local (frequent) dense patterns in their contact maps, and compile a library of such non- local interactions.

Asymptotic Enumeration of RNA Structures with Pseudoknots

A general framework for the computation of exponential growth rate and the asymptotic expansion for the numbers of k-noncrossing RNA structures with pseudoknots is developed and for arbitrary k singular expansions exist and via transfer theorems of analytic combinatorics, asymPTotic expression for the coefficients are obtained.