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Enumeration of Extended m-Regular Linear Stacks.

Qiang-Hui Guo1, Lisa H Sun1, Jian Wang1

  • 1Center for Combinatorics, LPMC-TJKLC, Nankai University , Tianjin, P.R. China .

Journal of Computational Biology : a Journal of Computational Molecular Cell Biology
|June 17, 2016
PubMed
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This study introduces extended linear stacks to model protein contact maps, showing their generating function relates to standard linear stacks. This allows deriving asymptotic formulas for protein fold structures.

Keywords:
$$m$$-regular linear stackcontact mapprotein foldzigzag stack

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Area of Science:

  • Computational Biology
  • Statistical Mechanics
  • Discrete Mathematics

Background:

  • Protein contact maps on 2D square lattices have specific constraints on arc length and vertex degrees.
  • Previous work enumerated [Formula: see text]-regular linear stacks with degree bounded by 2.

Purpose of the Study:

  • To study extended [Formula: see text]-regular linear stacks that better model real protein contact maps.
  • To establish a relationship between the generating functions of standard and extended linear stacks.

Main Methods:

  • Mathematical modeling using generating functions for linear stacks.
  • Derivation of a rational function relationship between generating functions.
  • Elimination of variables to obtain an equation for the extended generating function.

Main Results:

  • The generating function of extended [Formula: see text]-regular linear stacks ([Formula: see text]) is a rational function of the generating function for standard [Formula: see text]-regular linear stacks ([Formula: see text]).
  • An equation satisfied by [Formula: see text] was derived.
  • Asymptotic formulas for the number of [Formula: see text]-regular linear stacks were obtained.

Conclusions:

  • The extended linear stack model provides a closer approximation to real protein contact maps.
  • The established mathematical framework enables the asymptotic analysis of protein fold structures.