# Spatial Organization of Five-Fold Morphology as a Source of Geometrical Constraint in Biology

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Statistics of Spatial Organization for Shapes Γ

#### 2.2. Mathematical Basis of Eutacticity

#### 2.3. The Eutacticity and the Standard Deviation of Dispersion Mean of a Module

#### 2.4. Standard Deviation of Partition Variability

- Partitioning number (pn) defines the number of partitions inside a disc (ranging from three to ten).
- Partition variability (pv) determines multiple levels of variability (ten) inside discs by using random points, which in turn define the Voronoi diagrams. These levels of variability will be defined below.

- We initially consider a disk with a unitary radius where a second inscribed disk will be partitioned into a pn with a pv during the experiment (steps 4 and 5 of this algorithm, respectively). These discs are defined by particular features each: (a) The first disc is the external limit of the second and their coordinates are constant during the experiment; (b) the second one is constantly changing to obtain a pv (step 5 of this algorithm) and it is obtained by establishing a Voronoi tessellation. These two features (a) and (b) are described in the next steps (2) and (3) of this algorithm.
- Features of external disc. The boundaries of the external limit are defined by 24 fixed points generated as follows: The radius of the external disk is set to r = 1 and consecutive points are separated by an angle θ/24. The functionality of this feature lies in the establishment of a fixed limit of reference to maintain a constant area during variation of partitions.
- Features of internal disc. The boundaries of the internal limit are defined by 24 fixed points generated as follows: The radius of the internal disk is initially set to r = 0.53 ± 0.4 (established by the first level of variability step 6 of this algorithm) with 24 points consecutively separated by an angle θ/24. These radii are derived from a Voronoi tessellation whose points are the 24 points established before in this step besides the points derived from step 5. The functionality of this feature lies in the establishment of an internal limit able to change, providing statistical variation determining levels of variability of areas inside discs.
- Now, we define partition numbering (pn) inside the disk. Once the number of partitions is defined, say n (where 3 ≤ n ≤ 10 and $\mathrm{n}\in \mathbb{Z}$) to define a Voronoi tessellation, points are located in the disk at angles 2π/n ± 0.069 radians but at different radius. These radius values will define the pv, as described in the next item.
- Partition variability (pv). For each angular region defined above, 10 points are located at radius (between r = 0 and r = 10) at different positions to define different degrees of variability using Voronoi tessellations. The first point (first level of variability) is at r = 1. After the second point, all of them are located at random radius between 1 to 10. Hence, each level of variability (10) is given by radii ranges except 1 which is fixed at 1 (Figure 3); (a) 1, (b) 1–2, (c) 1–3, (d) 1–4, (e) 1–5, (f) 1–6, (g) 1–7, (h) 1–8, (i) 1–9 and (j) 1–10. Partition variability will define the broad spectrum of possibilities for area distribution inside discs without losing partitioning number. According to Equation (1), the average of areas requires a summation of sub-localities areas $\left({A}_{ij}\right)$ which were derived from partitions.
- Once the partition areas $\left({A}_{ij}\right)$ inside discs were obtained and Equation (1) was solved, Equation (2) is used to get standard deviations $\left({\sigma}_{i}\right)$ of variability for each disc. In order to normalize the level of variability for each pn, an index dividing the standard deviation of partitions and the particular area average of each partition was obtained (variability average; supporting information 2). There are eight particular area averages of partitions since we have a sample of 8 discs with different pn (from 3 to 10). These particular area averages are derived from a value n/(≈108.5 ± 1.5) which are n values obtained from the first level of variability (pv) at r = 1. It is important to say that the radius of the external disc (1) and the radius of the internal disc (r = 0.53 ± 0.4) was modified in order to get the particular area averages. However, in spite of the modification, the index between external discs and the internal ones remains constant. A sample of 20 discs to get 20 standard deviations $\left(20{\sigma}_{i}\right)$ was generated for each pn, and also for each level of pv (10) giving a sample of 200 discs for each pn. An average of standard deviations ($\overline{{\sigma}_{i}}$; variability average) was derived for each level of variability.Once the partition areas $\left({A}_{ij}\right)$ inside discs were obtained and Equation (1) was solved, Equation (2) is used to get standard deviations $\left({\sigma}_{i}\right)$ of variability for each disc. In order to normalize the level of variability for each pn, an index dividing the standard deviation of partitions and the particular area average of each partition was obtained (variability average; supporting information 2). There are eight particular area averages of partitions since we have a sample of 8 discs with different pn (from 3 to 10). These particular area averages are derived from a value n/(≈108.5 ± 1.5) which are n values obtained from the first level of variability (pv) at r = 1. It is important to say that the radius of the external disc (1) and the radius of the internal disc (r = 0.53 ± 0.4) was modified in order to get the particular area averages. However, in spite of the modification, the index between external discs and the internal ones remains constant. A sample of 20 discs to get 20 standard deviations $\left(20{\sigma}_{i}\right)$ was generated for each pn, and also for each level of pv (10) giving a sample of 200 discs for each pn. An average of standard deviations $(\overline{{\sigma}_{i}}$; variability average) was derived for each level of variability.
- Finally, a standard deviation of all variability averages is obtained for each pn.

## 3. Results

#### 3.1. Star Morphospace for Shapes Γ

#### 3.2. Experimental Evidence

## 4. Discussion

_{a}) remain in a zone with small SDM for modules, in contrast with those of a second set of stars with ε = 0.8 (ψ

_{b}). The morphospace of stars in Figure 5 shows that the mean eutactic value for stars with five vectors have a subtle major difference in contrast with stars with four and six vectors. Hence, two parameters are being important keys to detect spatial homogeneity inside morphospace, eutacticity and number of vectors. Whether dispersions of area distribution are associated with eutactic values we may conclude that small SDM for modules implies spatial homogeneity. Module variation implies fluctuating partitions without losing the particular correlation structure of the system related with polygonal side number. Therefore, the idea of modularity in our systems relies on a conserved polygonal structure even varying magnitudes of inner areas. One of our main hypotheses lies on considering these conserved polygonal structures as constraints that are not included as such, in our knowledge, in any other research. In that sense, polygons are changing in terms of spatial distribution depending on the number of vectors and the associated areas. Consequently, system-level properties for common architectures in simple polygonal forms are emergences of interacting spatial elements, attempting to gain space. One alternative, in terms of biological statistical mechanics, should be directed that endeavors as mechanical forces. However, according to our results, those forces might be considered as simple probabilistic parameters derived from standard deviations of modules immersed in polygons with a particular number of sides.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic properties of two different shapes Γ. (

**a**) A square is a locality associated to four subareas from four sub-localities ${S}_{1},{S}_{2},\dots ,{S}_{4}$ which are all equal; (

**b**) A shape Γ with a four-fold partition such that any of their sub-localities have unequal subareas is not regular; the set of areas defined by sub-localities ${S}_{1}$ and ${S}_{2}$ are smaller than those of ${S}_{3}$ and ${S}_{4}$.

**Figure 2.**Construction of a module from $k$ stars. A module is an average derived from an area summation of a particular sub-locality (e.g., sub-locality 1) from $k$ stars $\psi $ with a constant value $\epsilon $. In this figure, the second sub index of A is referring to sub-locality 1. Stars ${\psi}_{1},{\psi}_{2},\dots ,{\psi}_{k}$ are the building blocks to construct localities ${L}_{1},{L}_{2},\dots ,{L}_{k}$. This process is applied to build modules of the two experimental groups of stars, ${\psi}_{a}$ and ${\psi}_{b}$.

**Figure 3.**Defining partitioning number and partition variability. A disc is constructed to get Voronoi diagrams with constant area in spite of variability. The disc of this figure has a partitioning number of 2, one between axes x and y and the other is the remaining space. The magnitude of the radius defines ten levels of partition variability, which are the numbers emerging from the origin upon the diagonal on ray 3; (a) 1, (b) 1–2, (c) 1–3, (d) 1–4, (e) 1–5, (f) 1–6, (g) 1–7, (h) 1–8, (i) 1–9 and (j) 1–10. Each level of variability is given by radii ranges except (a) which is fixed at 1.

**Figure 4.**Dispersion mean of modules (DMM) and the standard variation of dispersion mean (SDM). DMM is the average of standard deviation of areas derived from Equation (9), from 100 localities using 100 sets of random points with several number of sub-localities with $\epsilon =1$ (${\psi}_{a}$; yellow bars) and $\epsilon =0.8$ (${\psi}_{b}$; orange bars). ANOVA test was performed in order to contrast eutactic values of DMM between ${\psi}_{a}$ and ${\psi}_{b}$. The obtained statistical significances of p range from less than 0.0001 for partitions with three modules and four modules (***); less than 0.05 for partitions with five modules (*); and less than 0.01 for partitions with six and seven modules (**). The null hypothesis was rejected in 23 of the 25 modules. The SDM (Equation (10)) for the module with $\epsilon =1$ (${\psi}_{a}$; grey bars) is notably smaller than the one obtained from the module with $\epsilon =0.8$ (${\psi}_{b}$; blue bars).

**Figure 5.**Star morphospace for eutacticity values derived from shapes Γ. Eutacticity means obtained from statistical distributions for vector stars ranging from 3 to 10 vectors.

**Figure 6.**Partitioning number and partition variation of planar discs. A sample of 40 planar discs shows how partitioning number (vertical left side) determines segmentation of an almost constant area (≈108.5 ± 1.5) into a particular number of sub-localities. Partition variability (bottom horizontal numbers) installs levels of variability giving 10 constant and subtle increases of area to generate random segmentations. Variability averages (right vertical graphics) reflect the average of standard deviations ($\overline{{\sigma}_{i}}$) which is derived for each level of variability. It is important to note how each increase of variability enhances heterogeneity for every partitioning equally even if the graphics are dissimilar. Partitioning number for discs with 7, 8, 9 and 10 regions are showed in Figure S5.

**Figure 7.**Standard deviation of all variability averages for each partitioning number. An average of standard deviations ($\overline{{\sigma}_{i}}$; variability average) was derived for each level of variability from Figure 6. A standard deviation of all variability averages is obtained for each partitioning number. According to this data, five-fold organizations are at the lowest level of dissimilarity among areas inside discs.

Set of Random Points ${\mathit{\omega}}_{\mathit{m},\mathit{n}}$ Defining the Associated Areas ${\mathit{A}}_{\mathit{i},\mathit{j}}$ for Sub-Locality 1 (Algorithm Defined in Reference [24]) | Summation of Areas for Star ${\mathit{\psi}}_{\mathit{k}}$ | |||||
---|---|---|---|---|---|---|

Stars | ${\omega}_{1,1}$ | ${\omega}_{1,2}$ | … | ${\omega}_{1,\alpha}$ | ||

${\psi}_{1}$ | ${A}_{1,1}^{{\omega}_{1,1}}$ | ${A}_{1,1}^{{\omega}_{1,2}}$ | … | ${A}_{1,1}^{{\omega}_{1,\alpha}}$ | ⇒ | $\frac{1}{\alpha}{\displaystyle {\displaystyle \sum}_{n=1}^{\alpha}}{A}_{1,1}^{{\omega}_{1,n}}$ |

${\psi}_{2}$ | ${A}_{2,1}^{{\omega}_{2,1}}$ | ${A}_{2,1}^{{\omega}_{2,2}}$ | … | ${A}_{2,1}^{{\omega}_{2,\alpha}}$ | ⇒ | $\frac{1}{\alpha}{\displaystyle {\displaystyle \sum}_{n=1}^{\alpha}}{A}_{2,1}^{{\omega}_{2,n}}$ |

. . . | . . . | . . . | . . . | . . . | . . . | . . . |

${\psi}_{k}$ | ${A}_{k,1}^{{\omega}_{k,1}}$ | ${A}_{k,1}^{{\omega}_{k,2}}$ | … | ${A}_{k,1}^{{\omega}_{k,\alpha}}$ | ⇒ | $\frac{1}{\alpha}{\displaystyle {\displaystyle \sum}_{n=1}^{\alpha}}{A}_{k,1}^{{\omega}_{k,n}}$ |

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**MDPI and ACS Style**

López-Sauceda, J.; López-Ortega, J.; Laguna Sánchez, G.A.; Sandoval Gutiérrez, J.; Rojas Meza, A.P.; Aragón, J.L.
Spatial Organization of Five-Fold Morphology as a Source of Geometrical Constraint in Biology. *Entropy* **2018**, *20*, 705.
https://doi.org/10.3390/e20090705

**AMA Style**

López-Sauceda J, López-Ortega J, Laguna Sánchez GA, Sandoval Gutiérrez J, Rojas Meza AP, Aragón JL.
Spatial Organization of Five-Fold Morphology as a Source of Geometrical Constraint in Biology. *Entropy*. 2018; 20(9):705.
https://doi.org/10.3390/e20090705

**Chicago/Turabian Style**

López-Sauceda, Juan, Jorge López-Ortega, Gerardo Abel Laguna Sánchez, Jacobo Sandoval Gutiérrez, Ana Paola Rojas Meza, and José Luis Aragón.
2018. "Spatial Organization of Five-Fold Morphology as a Source of Geometrical Constraint in Biology" *Entropy* 20, no. 9: 705.
https://doi.org/10.3390/e20090705