Fractal dimensions of neighborhoods

VR Veronica V. Rezelj LC Lucía Carrau FM Fernando Merwaiss LL Laura I. Levi DE Diana Erazo QT Quang Dinh Tran AH Annabelle Henrion-Lacritick VG Valérie Gausson YS Yasutsugu Suzuki DS Djoshkun Shengjuler BM Bjoern Meyer TV Thomas Vallet JW James Weger-Lucarelli VB Veronika Bernhauerová AT Avi Titievsky VS Vadim Sharov SP Stefano Pietropaoli MD Marco A. Diaz-Salinas VL Vincent Legros NP Nathalie Pardigon GB Giovanna Barba-Spaeth LB Leonid Brodsky MS Maria-Carla Saleh MV Marco Vignuzzi

This protocol is extracted from research article:

Defective viral genomes as therapeutic interfering particles against flavivirus infection in mammalian and mosquito hosts

**
Nat Commun**,
Apr 16, 2021;
DOI:
10.1038/s41467-021-22341-7

Defective viral genomes as therapeutic interfering particles against flavivirus infection in mammalian and mosquito hosts

Procedure

Fractal dimensions of neighborhoods were calculated for a more sensitive determination of the most enriched neighborhoods. A fractal dimension for a neighborhood of a center of radius ${r}_{t+1}$ with *t* + 1 closest distinct points in it can be calculated as follows. Let each point in the neighborhood be exchanged with the same neighborhood compressed with the $\frac{minimumDistance}{maximumDistance}\cong \frac{{r}_{1}}{2\cdot {r}_{t+1}}$ ratio. Next, each point in the compressed neighborhood *N*_{1} will be exchanged with the double compressed original neighborhood: compressed with the ${\left(\frac{{r}_{1}}{2\cdot {r}_{t+1}}\right)}^{2}$ ratio, next each point of the double compressed neighborhood will be exchanged with the triple compressed neighborhood: ${\left(\frac{{r}_{1}}{2\cdot {r}_{t+1}}\right)}^{3}$ ratio, and so on.

In this nested construction, balls of diminishing radiuses cover the infinite sequence of the nested compressed self-similar (fractal) ball-neighborhoods. Hence, the Hausdorff fractal dimension of the ball-neighborhood can be calculated as a minimal dimension *d* of a ball (volume without compression is ${r}_{t+1}^{d}$) in order to get the lowest estimation for sum of volumes of infinite series of all diminishing balls that cover all points in this nested fractal construction.

The Hausdorff content of this series is

Dimension *d* of this Hausdorff content of the ball-neighborhood is defined as:

Alternatively, the equivalent formula will be as follows:

This infimum for dimension is calculated as:

Indeed, for the above limit under $n\to \infty $ to be equal to zero the inequality as follows has to be true:

Taking the logarithm from both parts, the inequality will be transformed into equivalent:

Taking infimum we will get:

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

Note: The content above has been extracted from a research article, so it may not display correctly.

Q&A

Your question will be posted on the Bio-101 website. We will send your questions to the authors of this protocol and Bio-protocol community members who are experienced with this method. you will be informed using the email address associated with your Bio-protocol account.