Extended Topological Persistence and Contact Arrangements in Folded Linear Molecules

Sara Kališnik Verovšek,Alireza Mashaghi

doi:10.3389/fams.2016.00006

Sara Kališnik Verovšek, Alireza Mashaghi

Open Access

https://doi.org/10.3389/fams.2016.00006

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Abstract

Structure plays a pivotal role in determining the functional properties of self-interacting linear biomolecular chains, for example proteins and nucleic acids. In this paper, we propose a method for representing each such molecule combinatorially - as a one-dimensional simplicial complex - in a novel way that takes into account intra-chain contacts. The representation allows for efficient quantification of structural similarities and differences between molecules, and for studying molecular topology using extended persistence. This method performs a multi-scale analysis on a filtered simplicial complex as it tracks clusters, holes, and higher dimensional voids in the filtration. From extended persistence we extract information about the arrangement of intra-chain interactions, a topological property which demonstrably affects folding and unfolding dynamics of the linear chains.

Highlights

Proteins and nucleic acids are linear organic polymers which are present in and vital to every living cell [1, 2]
Topology is the branch of mathematics which deals with shape
We first use the algebraic tools presented in the previous section to model the molecular chains with the most basic circuit topologies as simplicial complexes (Figure 4)

Summary

Introduction

Proteins and nucleic acids are linear organic polymers which are present in and vital to every living cell [1, 2]. They perform a vast array of functions within living organisms, for example, proteins provide mechanical support of cells, catalyze specific chemical reactions as well as transport and store nutrients and metabolites. In determining the functional properties of proteins and nucleic acids, and more generally, self-interacting molecular chains, shape plays a pivotal role. Chemists have been aware of the connection between the shape of a molecule and its function for decades already. Topological ideas have inspired methods for visualizing complex datasets [13]

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