# Statistical Properties of Lasso-Shape Polymers and Their Implications for Complex Lasso Proteins Function

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Random Lassos Generation

#### 2.2. Simulation Model

#### 2.3. Data Analysis

#### 2.4. Proteins Analyzed

#### 2.5. Graphics

## 3. Results

#### 3.1. Probability That a Lasso Is Complex

#### 3.2. Shape Parameters of a Lasso Loop

#### 3.3. Comparison Of Simulated Polymers with Complex Lasso Proteins

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Complex lasso proteins.(

**A**) An exemplary complex lasso protein (PDB code 2mxq) with one piercing through the loop (${L}_{1}$ type). The orange beads denote the cysteine residues; (

**B**) Schematic depiction of six complex lasso motifs, with the orange bar denoting the bridge.

**Figure 2.**Description of the model and shape parameters. (

**A**) The schematic presentation of the model used in the simulation–the shape of the freely moving green loop was analyzed; (

**B**) The polymer with its ellipsoid of inertia (top) and the semi-axis of the ellipsoid used to calculate the shape parameters (bottom); (

**C**) The exemplary ellipsoids with their location on the contour plot of the asphericity (orange curves) and prolateness (blue curves) in the space of the fraction of ellipsoid semi-axes $a>b>c$. Note that for spherical ellipsoid ($b/a=c/a=1$), the prolateness is undetermined.

**Figure 3.**The probability of trivial lassos. (

**A**) A plot of the probability for an equal length of the tail t and the loop N. In the inset, the schematic structure of trivial lasso; (

**B**) Probability of trivial lasso vs tail length t for fixed loop length N; (

**C**) Probability of trivial lasso vs loop length N for fixed tail length t; (

**D**) The surface of trivial lasso probability in the space of loop and tail length. In the case of (

**B**) and (

**C**) only selected traces were plotted to maintain transparency. For all the data see Section S2 of Supplementary Materials.

**Figure 4.**Analysis of non-trivial lassos. (

**Left panel**) The probability for a given number of piercings as a function of the loop size for an equal loop size N and tail length t for non-trivial lassos only. “Other” means lassos with ≥ 5 piercings.; (

**Right panel**) The domination number d as a function of the loop length N and tail length $t=N$ for non-trivial lassos. The domination number d is a maximal number of piercings p for which lassos with $\ge p$ piercings are the majority of non-trivial lassos with given N and t (Equation (3)).

**Figure 5.**The dependence of shape parameters on the loop length and thread thickness. (

**A**) Radius of gyration; (

**B**) Distension; (

**C**) Asphericity; (

**D**) Prolateness; (

**E**) The trajectories of the loop shapes in the space of the fraction of ellipsoid semi-axes $a>b>c$. The traces start from the loops with at least 8 beads. The contour lines of asphericity (orange) and prolateness (blue) were added. Note that the thread thickness determines the smallest possible loop (for details see Materials and Methods).

**Figure 6.**The complex lasso shapes in proteins. (

**A**) The probability of trivial protein lasso versus the trivial lasso polymer surface, as a function of loop and tail length. (

**B**) The expected and observed number of the non-trivial lassos as a function of the chain length. The light color curves show the smoothed traces. The normalized histogram of (

**C**) loop lengths and (

**D**) asphericity for the pierced and non-threaded (trivial) case. The arrows denote the shift of the distribution from trivial to the non-trivial case. The dotted lines denote the smoothed fit.

**Table 1.**Parameters fitted for the shape parameters scaling as the function of the thread thickness. The “ratio” is the value of the preexponential factor compared with the case of unthreaded loop ${a}_{r}(thickness=n)/{a}_{r}(thickness=0)$. The scaling factor for the asphericity for the unthreaded loop was given in italic as it stands out from the trend and the fitting error obtained in this case was significant (see Section S3 in Supplementary Materials).

Thread | Radius of Gyration | Asphericity | Prolatness | |||||
---|---|---|---|---|---|---|---|---|

Thickness | ${\mathit{a}}_{\mathit{R}}\left[\mathit{nm}\right]$ | ${\mathit{c}}_{\mathit{R}}\left[\mathit{nm}\right]$ | ratio | ${\mathit{a}}_{\mathit{A}}$ | $\mathit{\mu}$ | ${\mathit{A}}_{\mathit{\infty}}$ | ${\mathit{a}}_{\mathit{P}}$ | ${\mathit{P}}_{\mathit{\infty}}$ |

0 | 0.154 | −0.102 | 1.00 | 0.0342 | −0.186 | 0.0708 | −1.37 | 0.300 |

1 | 0.164 | −0.0825 | 1.06 | 0.188 | −0.753 | 0.0853 | −3.07 | 0.315 |

2 | 0.164 | −0.0162 | 1.06 | 0.353 | −0.822 | 0.0858 | −4.57 | 0.384 |

3 | 0.164 | 0.0376 | 1.06 | 0.845 | −0.993 | 0.0869 | −5.53 | 0.420 |

4 | 0.164 | 0.106 | 1.06 | 1.05 | −0.969 | 0.0856 | −6.10 | 0.413 |

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Dabrowski-Tumanski, P.; Gren, B.; Sulkowska, J.I.
Statistical Properties of Lasso-Shape Polymers and Their Implications for Complex Lasso Proteins Function. *Polymers* **2019**, *11*, 707.
https://doi.org/10.3390/polym11040707

**AMA Style**

Dabrowski-Tumanski P, Gren B, Sulkowska JI.
Statistical Properties of Lasso-Shape Polymers and Their Implications for Complex Lasso Proteins Function. *Polymers*. 2019; 11(4):707.
https://doi.org/10.3390/polym11040707

**Chicago/Turabian Style**

Dabrowski-Tumanski, Pawel, Bartosz Gren, and Joanna I. Sulkowska.
2019. "Statistical Properties of Lasso-Shape Polymers and Their Implications for Complex Lasso Proteins Function" *Polymers* 11, no. 4: 707.
https://doi.org/10.3390/polym11040707