# Application of Lacunarity for Quantification of Single Molecule Localization Microscopy Images

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Lacunarity Calculation

^{2}size box at every possible different position on the image. The mass of these boxes is calculated by counting the number of object pixels inside them. Lacunarity can then be calculated from the sums of the first and second moments of the box masses. As we previously mentioned, in 2D SMLM the data consist of coordinate pairs, wherefore the calculation of lacunarity must be modified accordingly. Figure 1a shows a regular SMLM dataset consisting of numerous coordinate pairs in an L by L area where “L” is the side length of the region of interest (ROI) in nanometers. To calculate lacunarity we need to redefine the mass of a box as the number of localizations inside the box. We also need to define a step size “s” with which the boxes will glide through the image. As a consequence, the smallest possible box size will be equal to “s”. This gliding can be seen in Figure 1b. As the boxes glide through the image, one can calculate the mass of each box and write the value into a matrix “BM(i,j)” as shown in Figure 1c. To speed up the calculations, the SMLM data can be pixelized into an M by M image, where

#### 2.2. TestSTORM Simulation

^{3})” and the “Number of frames” were changed. No sample drift was introduced and a Gaussian PSF was used. Each blinking event was fitted with the rainSTORM reconstruction software using the multi-Gaussian 2D analysis algorithm. The following sample parameters were evaluated at five values: the number of clusters defined as the number of clusters in the region of interest, the size of clusters defined as the radius of a single cluster, the distance of clusters defined as the distance between the center of two neighbouring clusters, the density of nanofoci defined as the number of nanofoci over a square micron area inside of the clusters, the size of nanofoci defined as the radius of a single nanofocus, the number of localizations per nanofoci defined as the number of fluorophores belonging to a single nanofocus, and nonspecific localization density defined as the number of nonspecific localizations in a cube micron volume.

#### 2.3. DNA DSB dSTORM Images

## 3. Results

#### 3.1. Lacunarity Behaviour Examined through TestSTORM Simulations

^{2}, 25 nanofoci/μm

^{2}, 40 nanofoci/μm

^{2}, 55 nanofoci/μm

^{2}and 70 nanofoci/μm

^{2}. The super-resolution images for 10/μm

^{2}, 40/μm

^{2}and 70/μm

^{2}nanofocus densities can be seen in Figure 3a–c. An increase in the density of nanofoci increases the number of localizations inside the clusters. The localization density increases from 3800 localizations/μm

^{2}to 7200 localizations/μm

^{2}, 9700 localizations/μm

^{2}, 11,300 localizations/μm

^{2}and 12,500 localizations/μm

^{2}, creating a more even distribution. This effect results in an increase in homogeneity to a certain point determined by the cluster size and shape. The peak of the lacunarity difference curve slightly shifts (69 nm → 78 nm) towards the larger box sizes, indicating that the homogenization effect is greater for the smaller box sizes (Figure 3j). This is explained by the fact that the introduction of new nanofoci only affects the image in a small localized area.

^{3}, 70/μm

^{3}, 140/μm

^{3}, 280/μm

^{3}and 560/μm

^{3}. The super-resolution images for 0/μm

^{3}, 140/μm

^{3}and 560/μm

^{3}nonspecific localization densities can be seen in Figure 4a–c. Higher densities of nonspecific localizations evenly homogenize the whole image, while the peak of the curve only moves very slightly (69 nm → 64 nm) towards the smaller box sizes (Figure 4d).

#### 3.2. Dose Dependent Lacunarity Study of the X-ray Radiation Treated Cell Line

#### 3.3. Kinetic Lacunarity Study of the X-ray Radiation Treated Cell Line

#### 3.4. Lacunarity Study of Chemically Treated Cell Lines

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Step by step representation of the lacunarity calculation process and visualization. Localization cluster of a hundred localizations in a Gaussian distribution of 150 nm sigma on an area of 1000 nm by 1000 nm (

**a**). Zoomed in on the cluster showing the gliding of an “ε” sized box at a step size of “s” (

**b**). The box masses in the zoomed in area, calculated from the number of localizations in each box (

**c**). Random set of a hundred localizations on the same area of 1000 nm by 1000 nm with an even distribution (

**d**). Lacunarity curves of the cluster and the random dataset (

**e**). Lacunarity difference curve created by calculating the relative difference between the lacunarity curve of the cluster and that of the random dataset of the same number of localizations at each point (

**f**).

**Figure 2.**Effects of changing cluster parameters on lacunarity. Lacunarity difference curves of TestSTORM generated datasets of different cluster numbers (

**a**), cluster sizes (

**b**), cluster distances (

**c**). Three super-resolution images of the simulated data are also shown for each lacunarity difference curve. Cluster numbers of one (

**d**), five (

**e**) and nine (

**f**). Cluster sizes of 140 nm (

**g**), 560 nm (

**h**) and 980 nm (

**i**). Cluster distances of 500 nm (

**j**), 1500 nm (

**k**) and 2500 nm (

**l**). Scale bars are 1 µm.

**Figure 3.**Effects of changing nanofocus parameters on lacunarity. Lacunarity difference curves of TestSTORM generated datasets of different nanofocus densities (

**a**), nanfocus sizes (

**b**), localizations per nanofocus (

**c**). Three super-resolution images of the simulated data are also shown for each lacunarity difference curve. Nanofocus densities of 10/μm

^{2}(

**d**), 40/μm

^{2}(

**e**) and 70/μm

^{2}(

**f**). Nanofocus sizes of 11 nm (

**g**), 55 nm (

**h**) and 99 nm (

**i**). Number of localizations per nanofocus of 10 (

**j**), 150 (

**k**) and 290 (

**l**). Scale bars are 1 µm.

**Figure 4.**Effects of changing nonspecific localization densities on lacunarity and summary of the simulation results. Lacunarity difference curves of TestSTORM generated datasets of different nonspecific localization densities (

**a**). Three super-resolution images of the simulated data with nonspecific localization densities of 0/μm

^{3}(

**b**), 140/μm

^{3}(

**c**) and 560/μm

^{3}(

**d**) are also shown. Scale bars are 1 µm. The peak positions for each simulation in order (

**e**). Number of clusters (NC), cluster size (CS), cluster distance (CD), nanofocus density (ND), nanofocus size (NS), localizations per nanofocus (LPN) and nonspecific localization density (NSL).

**Figure 5.**Super-resolution images and lacunarity difference curves of Alexa Fluor 647 labelled γH2AX in the nuclei of U2OS cells grouped by radiation dosage. Each cell was observed 30 min after being subjected to 0 Grays (

**a**,

**b**), 2 Grays (

**c**,

**d**) and 5 Grays (

**e**,

**f**) of ionizing radiation. Scale bars are 1 µm. The number of studied cells is 6, 7 and 9, respectively. On each lacunarity difference curve, the average of the curves for each cell is shown in red and the average of the untreated U2OS is shown in dashed blue.

**Figure 6.**Super-resolution images and lacunarity difference curves of A647 labelled γH2AX in the nuclei of U2OS cells grouped by time after treatment. The cells were observed 30 min (

**a**,

**b**), 24 h (

**c**,

**d**) and 72 h (

**e**,

**f**) after being subjected to 5 Grays of ionizing radiation. Scale bars are 1 µm. The number of studied cells is 9, 6 and 5, respectively. On each lacunarity difference curve, the average of the curves for each cell is shown in red and the average of the untreated U2OS is shown in dashed blue.

**Figure 7.**Super-resolution images and lacunarity difference curves of A647 labelled γH2AX in the nuclei of U2OS and DIvA cells grouped by treatment. Untreated U2OS (

**a**,

**b**), NCS treated U2OS (

**c**,

**d**) and 4-OHT treated DIvA (

**e**,

**f**) cells. Scale bars are 1 µm. The number of studied cells is four, five, and six, respectively. On each lacunarity difference curve, the average of the curves for each cell is shown in red and the average of the untreated U2OS is shown in dashed blue.

Names of the Simulation Parameters | Base Values of the Simulation Parameters | Ranges of the Simulation Parameters |
---|---|---|

Cluster number | 9 | 1–9 |

Cluster size (nm) | 560 | 140–980 |

Cluster distance (nm) | 2500 | 500–2500 |

Nanofocus density (nanofoci/µm^{2}) | 40 | 10–70 |

Nanofocus Size (nm) | 55 | 11–99 |

Localizations per nanofocus (localizations/µm^{2}) | 150 | 10–290 |

Nonspecific localization density (localizations/µm^{3}) | 70 | 0–560 |

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**MDPI and ACS Style**

Kovács, B.B.H.; Varga, D.; Sebők, D.; Majoros, H.; Polanek, R.; Pankotai, T.; Hideghéty, K.; Kukovecz, Á.; Erdélyi, M. Application of Lacunarity for Quantification of Single Molecule Localization Microscopy Images. *Cells* **2022**, *11*, 3105.
https://doi.org/10.3390/cells11193105

**AMA Style**

Kovács BBH, Varga D, Sebők D, Majoros H, Polanek R, Pankotai T, Hideghéty K, Kukovecz Á, Erdélyi M. Application of Lacunarity for Quantification of Single Molecule Localization Microscopy Images. *Cells*. 2022; 11(19):3105.
https://doi.org/10.3390/cells11193105

**Chicago/Turabian Style**

Kovács, Bálint Barna H., Dániel Varga, Dániel Sebők, Hajnalka Majoros, Róbert Polanek, Tibor Pankotai, Katalin Hideghéty, Ákos Kukovecz, and Miklós Erdélyi. 2022. "Application of Lacunarity for Quantification of Single Molecule Localization Microscopy Images" *Cells* 11, no. 19: 3105.
https://doi.org/10.3390/cells11193105