# Topological Models for Open-Knotted Protein Chains Using the Concepts of Knotoids and Bonded Knotoids

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## Abstract

**:**

## 1. Introduction

## 2. Knotoids

## 3. Results

#### 3.1. Analyzing Open Protein Chains Using Knotoids

#### 3.2. A Topological Model for Bonded Open Protein Chains

#### An Application to Complex Lassos

## 4. Discussion

## 5. Materials and Methods

#### 5.1. Construction of Projection Globes and Maps

#### 5.2. The Turaev Loop Bracket

#### 5.3. The Arrow Polynomial

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

KMT algorithm | Koniaris, Muthukumar, Taylor algorithm |

PDB | Protein Database |

## References

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**Figure 1.**Various types of knotoids. A knot-type knotoid (

**a**); a proper knotoid (

**b**); and a non-trivial planar knotoid (

**c**).

**Figure 3.**The planar knotoid $k2.{3}^{p}$ is trivial when considered in ${S}^{2}$. We start by placing $k2.{3}^{p}$ in ${S}^{2}$ (

**a**). Then, the lower arc is stretched (

**b**) and pulled towards and past the south pole, around the back (

**c**) and finally is brought up again (

**d**). The application of two Reidemeister I moves completes the proof (

**e,f**). The dotted lines indicate parts of the diagram that are on the back side of the sphere.

**Figure 4.**The projection globes for (

**a**) the uniform closure technique; (

**b**) the spherical knotoids technique; and (

**c**) the planar knotoids technique.

**Figure 5.**The projection maps for (

**a**) the uniform closure technique; (

**b**) the spherical knotoids technique; and (

**c**) the planar knotoids technique.

**Figure 7.**Type D (

**a**,

**b**), and type C (

**c**,

**d**) substitutions. Type D substitutions distinguish (

**e**) sequential, (

**f**) nested and (

**g**) pseudoknot-like bonds by applying to the substitution a polynomial invariant for knotoids like the Turaev loop bracket polynomial or the arrow polynomial.

**Figure 8.**The protein 2LFK as (

**a**) a cartoon; and (

**b**) a polygonal curve. Same colored spheres indicate that there is a bond between them. More precisely, the green dots correspond to the pair of residues with indices 24 and 51, while the red to the pair 52 and 69. Below (

**c**) is the corresponding bonded knotoid diagram and the appropriate substitutions. In the diagram the bonds are represented by colored dashed lines.

**Figure 9.**Distinguishing the lassos (

**a**) ${L}_{0}$; (

**b**) ${L}_{1}$; (

**c**) ${L}_{2}$; (

**d**) $L{L}_{1,1}$; and (

**e**) ${L}_{s}$. The crossing the determines the first piercing of the loop is positive and so a type $D+$ insertion is performed on the lassos. The slipknot in ${L}_{2}$ is detected through progressive trimming of the C terminus of the three-dimensional curve and evaluating each resulting diagram, while the first two lassos are distinguished immediately by the insertion. Insertions in lassos (

**d**,

**e**) distinguish them from the types that were already discussed, but not from one another. Trimming the C terminus of the 3D chain leads to different substructures allowing, thus, their distinction.

**Figure 10.**The Turaev loop bracket rules. (

**a**,

**b**) The smoothing rules; (

**c**) the loop value; (

**d**) the value for nested loops enclosing a long segment; (

**e**) the long segment value.

**Figure 11.**(

**a**) Oriented state expansion; (

**b**) Reduction rules for the arrow polynomial; and (

**c**) a component with one (${\mathsf{\Lambda}}_{1}$) and two zig-zags (${\mathsf{\Lambda}}_{2}$), respectively.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Goundaroulis, D.; Gügümcü, N.; Lambropoulou, S.; Dorier, J.; Stasiak, A.; Kauffman, L. Topological Models for Open-Knotted Protein Chains Using the Concepts of Knotoids and Bonded Knotoids. *Polymers* **2017**, *9*, 444.
https://doi.org/10.3390/polym9090444

**AMA Style**

Goundaroulis D, Gügümcü N, Lambropoulou S, Dorier J, Stasiak A, Kauffman L. Topological Models for Open-Knotted Protein Chains Using the Concepts of Knotoids and Bonded Knotoids. *Polymers*. 2017; 9(9):444.
https://doi.org/10.3390/polym9090444

**Chicago/Turabian Style**

Goundaroulis, Dimos, Neslihan Gügümcü, Sofia Lambropoulou, Julien Dorier, Andrzej Stasiak, and Louis Kauffman. 2017. "Topological Models for Open-Knotted Protein Chains Using the Concepts of Knotoids and Bonded Knotoids" *Polymers* 9, no. 9: 444.
https://doi.org/10.3390/polym9090444