Topological balance of cell distributions in plane monolayers

Most of normal proliferative epithelia of plants and metazoans are topologically invariant and characterized by similar cell distributions according to the number of cell neighbors (DCNs). Here we study peculiarities of these distributions and explain why the DCN obtained from the location of intercellular boundaries and that based on the Voronoi tessellation with nodes located on cell nuclei may differ from each other. As we demonstrate, special microdomains where four or more intercellular boundaries converge are topologically charged. Using this fact, we deduce a new equation describing the topological balance of the DCNs. The developed theory is applied for a series of microphotographs of non-tumoral epithelial cells of the human cervix (HCerEpiC) to improve the image processing near the edges of microphotographs and reveal the topological invariance of the examined monolayers. Special contact microdomains may be present in epithelia of various natures, however, considering the well-known vertex model of epithelium, we show that such contacts are absent in the usual solid-like state of the model and appear only in the liquid-like cancer state. Also, we discuss a possible biological role of special contacts in context of proliferative epithelium dynamics and tissue morphogenesis.

An improved trabecular bone model based on Voronoi tessellation

BACKGROUND AND OBJECTIVE

Accurate numerical and physical models of trabecular bone, correctly representing its complexity and variability, could be highly advantageous in the development of e.g. new bone-anchored implants due to the limited availability of real bone. Several Voronoi tessellation-based porous models have been reported in the literature, attempting to mimic the trabecular bone. However, these models have been limited to lattice rod-like structures, which are only structurally representative of very high-porosity trabecular bone. The objective of this study was to provide an improved model, more representative of trabecular bone of different porosity.

METHODS

Boolean operations were utilized to merge scaled Voronoi cells, thereby introducing different structural patterns, controlling porosity and to some extent anisotropy. The mechanical properties of the structures were evaluated using analytical estimations, numerical simulations, and experimental compression tests of 3D-printed versions of the structures. The capacity of the developed models to represent trabecular bone was assessed by comparing some key geometric features with trabecular bone characterized in previous studies.

RESULTS

The models gave the possibility to provide pore interconnectivity at relatively low porosities as well as both plate- and rod-like structures. The mechanical properties of the generated models were predictable with numerical simulations as well as an analytical approach. The permeability was found to be better than Sawbones at the same porosity. The models also showed the capability of matching e.g. some vertebral structures for key geometric features.

CONCLUSIONS

An improved numerical model for mimicking trabecular bone structures was successfully developed using Voronoi tessellation and Boolean operations. This is expected to benefit both computational and experimental studies by providing a more diverse and representative structure of trabecular bone.

VoroIF-GNN: Voronoi tessellation-derived protein-protein interface assessment using a graph neural network

We present VoroIF-GNN (Voronoi InterFace Graph Neural Network), a novel method for assessing inter-subunit interfaces in a structural model of a protein-protein complex, relying solely on the input structure without any additional information. Given a multimeric protein structural model, we derive interface contacts from the Voronoi tessellation of atomic balls, construct a graph of those contacts, and predict the accuracy of every contact using an attention-based GNN. The contact-level predictions are then summarized to produce whole interface-level scores. VoroIF-GNN was blindly tested for its ability to estimate the accuracy of protein complexes during CASP15 and showed strong performance in selecting the best multimeric model out of many. The method implementation is freely available at https://kliment-olechnovic.github.io/voronota/expansion_js/.

A mechanistic computational framework to investigate the hemodynamic fingerprint of the blood oxygenation level-dependent signal

Blood oxygenation level-dependent (BOLD) functional magnetic resonance imaging (fMRI) is one of the most used imaging techniques to map brain activity or to obtain clinical information about human cortical vasculature, in both healthy and disease conditions. Nevertheless, BOLD fMRI is an indirect measurement of brain functioning triggered by neurovascular coupling. The origin of the BOLD signal is quite complex, and the signal formation thus depends, among other factors, on the topology of the cortical vasculature and the associated hemodynamic changes. To understand the hemodynamic evolution of the BOLD signal response in humans, it is beneficial to have a computational framework available that virtually resembles the human cortical vasculature, and simulates hemodynamic changes and corresponding MRI signal changes via interactions of intrinsic biophysical and magnetic properties of the tissues. To this end, we have developed a mechanistic computational framework that simulates the hemodynamic fingerprint of the BOLD signal based on a statistically defined, three-dimensional, vascular model that approaches the human cortical vascular architecture. The microvasculature is approximated through a Voronoi tessellation method and the macrovasculature is adapted from two-photon microscopy mice data. Using this computational framework, we simulated hemodynamic changes-cerebral blood flow, cerebral blood volume, and blood oxygen saturation-induced by virtual arterial dilation. Then we computed local magnetic field disturbances generated by the vascular topology and the corresponding blood oxygen saturation changes. This mechanistic computational framework also considers the intrinsic biophysical and magnetic properties of nearby tissue, such as water diffusion and relaxation properties, resulting in a dynamic BOLD signal response. The proposed mechanistic computational framework provides an integrated biophysical model that can offer better insights regarding the spatial and temporal properties of the BOLD signal changes.

Shannon Entropy of Ramsey Graphs with up to Six Vertices

Shannon entropy quantifying bi-colored Ramsey complete graphs is introduced and calculated for complete graphs containing up to six vertices. Complete graphs in which vertices are connected with two types of links, labeled as α-links and β-links, are considered. Shannon entropy is introduced according to the classical Shannon formula considering the fractions of monochromatic convex α-colored polygons with n α-sides or edges, and the fraction of monochromatic β-colored convex polygons with m β-sides in the given complete graph. The introduced Shannon entropy is insensitive to the exact shape of the polygons, but it is sensitive to the distribution of monochromatic polygons in a given complete graph. The introduced Shannon entropies Sα and Sβ are interpreted as follows: Sα is interpreted as an average uncertainty to find the green α-polygon in the given graph; Sβ is, in turn, an average uncertainty to find the red β-polygon in the same graph. The re-shaping of the Ramsey theorem in terms of the Shannon entropy is suggested. Generalization for multi-colored complete graphs is proposed. Various measures quantifying the Shannon entropy of the entire complete bi-colored graphs are suggested. Physical interpretations of the suggested Shannon entropies are discussed.

Co-cultures of cerebellar slices from mice with different reelin genetic backgrounds as a model to study cortical lamination

Background: Reelin has fundamental functions in the developing and mature brain. Its absence gives rise to the Reeler phenotype in mice, the first described cerebellar mutation. In homozygous mutants missing the Reelin gene ( reln -/-), neurons are incapable of correctly positioning themselves in layered brain areas such as the cerebral and cerebellar cortices. We here demonstrate that by employing ex vivo cultured cerebellar slices one can reduce the number of animals and use a non-recovery procedure to analyze the effects of Reelin on the migration of Purkinje neurons (PNs). Methods: We generated mouse hybrids (L7-GFP relnF1/) with green fluorescent protein (GFP)-tagged PNs, directly visible under fluorescence microscopy. We then cultured the slices obtained from mice with different reln genotypes and demonstrated that when the slices from reln -/- mutants were co-cultured with those from reln +/- mice, the Reelin produced by the latter induced migration of the PNs to partially rescue the normal layered cortical histology. We have confirmed this observation with Voronoi tessellation to analyze PN dispersion. Results: In images of the co-cultured slices from reln -/- mice, Voronoi polygons were larger than in single-cultured slices of the same genetic background but smaller than those generated from slices of reln +/- animals. The mean roundness factor, area disorder, and roundness factor homogeneity were different when slices from reln -/- mice were cultivated singularly or co-cultivated, supporting mathematically the transition from the clustered organization of the PNs in the absence of Reelin to a layered structure when the protein is supplied ex vivo. Conclusions: Neurobiologists are the primary target users of this 3Rs approach. They should adopt it for the possibility to study and manipulate ex vivo the activity of a brain-secreted or genetically engineered protein (scientific perspective), the potential reduction (up to 20%) of the animals used, and the total avoidance of severe surgery (3Rs perspective).

Co-cultures of cerebellar slices from mice with different reelin genetic backgrounds as a model to study cortical lamination

Background: Reelin has fundamental functions in the developing and mature brain. Its absence gives rise to the Reeler phenotype in mice, the first described cerebellar mutation. In homozygous mutants missing the Reelin gene ( reln -/-), neurons are incapable of correctly positioning themselves in layered brain areas such as the cerebral and cerebellar cortices. We here demonstrate that by employing ex vivo cultured cerebellar slices one can reduce the number of animals and use a non-recovery procedure to analyze the effects of Reelin on the migration of Purkinje neurons (PNs). Methods: We generated mouse hybrids (L7-GFP relnF1/) with green fluorescent protein (GFP)-tagged PNs, directly visible under fluorescence microscopy. We then cultured the slices obtained from mice with different reln genotypes and demonstrated that when the slices from reln -/- mutants were co-cultured with those from reln +/- mice, the Reelin produced by the latter induced migration of the PNs to partially rescue the normal layered cortical histology. We have confirmed this observation with Voronoi tessellation to analyze PN dispersion. Results: In images of the co-cultured slices from reln -/- mice, Voronoi polygons were larger than in single-cultured slices of the same genetic background but smaller than those generated from slices of reln +/- animals. The mean roundness factor, area disorder, and roundness factor homogeneity were different when slices from reln -/- mice were cultivated singularly or co-cultivated, supporting mathematically the transition from the clustered organization of the PNs in the absence of Reelin to a layered structure when the protein is supplied ex vivo. Conclusions: Neurobiologists are the primary target users of this 3Rs approach. They should adopt it for the possibility to study and manipulate ex vivo the activity of a brain-secreted or genetically engineered protein (scientific perspective), the potential reduction (up to 20%) of the animals used, and the total avoidance of severe surgery (3Rs perspective).

True-to-scale DNA-density maps correlate with major accessibility differences between active and inactive chromatin

Chromatin compaction differences may have a strong impact on accessibility of individual macromolecules and macromolecular assemblies to their DNA target sites. Estimates based on fluorescence microscopy with conventional resolution, however, suggest only modest compaction differences (∼2-10×) between the active nuclear compartment (ANC) and inactive nuclear compartment (INC). Here, we present maps of nuclear landscapes with true-to-scale DNA densities, ranging from <5 to >300 Mbp/μm3. Maps are generated from individual human and mouse cell nuclei with single-molecule localization microscopy at ∼20 nm lateral and ∼100 nm axial optical resolution and are supplemented by electron spectroscopic imaging. Microinjection of fluorescent nanobeads with sizes corresponding to macromolecular assemblies for transcription into nuclei of living cells demonstrates their localization and movements within the ANC and exclusion from the INC.

Voronoi Tessellations and the Shannon Entropy of the Pentagonal Tilings

We used the complete set of convex pentagons to enable filing the plane without any overlaps or gaps (including the Marjorie Rice tiles) as generators of Voronoi tessellations. Shannon entropy of the tessellations was calculated. Some of the basic mosaics are flexible and give rise to a diversity of Voronoi tessellations. The Shannon entropy of these tessellations varied in a broad range. Voronoi tessellation, emerging from the basic pentagonal tiling built from hexagons only, was revealed (the Shannon entropy of this tiling is zero). Decagons and hendecagon did not appear in the studied Voronoi diagrams. The most abundant Voronoi tessellations are built from three different kinds of polygons. The most widespread is the combination of pentagons, hexagons, and heptagons. The most abundant polygons are pentagons and hexagons. No Voronoi tiling built only of pentagons was registered. Flexible basic pentagonal mosaics give rise to a diversity of Voronoi tessellations, which are characterized by the same symmetry group. However, the coordination number of the vertices is variable. These Voronoi tessellations may be useful for the interpretation of the iso-symmetrical phase transitions.

Simulation of Diffusion-Controlled Growth of Interdependent Nuclei under Potentiostatic Conditions

The problem of diffusion-controlled growth following an instantaneous nucleation event was studied within the framework of a new numerical model, considering the spatial distribution of hemispherical nuclei on the electrode surface and the mutual influence of growing nuclei via the collision of 3D diffusion fields. The simulation of the diffusion-controlled growth of hexagonal and random ensembles was performed at the overpotential-dependent number density of nuclei. The diffusion flow to each nucleus within a random ensemble was simulated by the finite difference method using the derived analytical expressions for the surface areas and the volumes formed at the intersection of 3D diffusion fields with the side faces of a virtual right prism with a Voronoi polygon base. The implementation of this approach provides an accurate calculation of concentration profiles, time dependences of the size of nuclei, and current transients. The results, including total current density transients, growth exponents, and nucleus size distribution, were compared with models developed within the concept of planar diffusion zones, the mean-field approximation and the Brownian dynamics simulation method, as well as with experimental data from the literature. The prospects of the model for studying the initial stages of electrocrystallization were discussed.

Additively Manufactured Multilevel Voronoi-Lattice Scaffolds with Bonelike Mechanical Properties

Irregular porous scaffold through Voronoi tessellation based on global modeling demonstrated randomness to a certain degree and susceptibility to producing large processing deviations. A modeling method for new types of scaffolds based on periodic arrays of Voronoi unit cell was proposed in this study. These porous scaffolds presented controllable local cells and satisfactory mechanical properties. The topological structure of the Voronoi unit cell was controlled using three independent cell design factors (Voronoi polyhedron volume V, face-centered scaled factor F₁, and body-centered scaled factor F₂), and multilevel Voronoi-lattice scaffolds were constructed on the basis of periodic arrays of the Voronoi unit cell. Compressive test and simulation were combined to quantify the mechanical properties of scaffolds. The regression equations were established using the response surface method (RSM) to determine relationships between Voronoi unit cell design factors and structural characteristic parameters and mechanical properties. The same trends were observed in stress-strain curves of the compressive test and simulation. The mechanical properties of scaffolds can be appropriately quantified via simulation. Regression equations based on RSM can properly predict the structural characteristic parameters and mechanical properties of the scaffold. Compared with V, F₁ and F₂ exerted a stronger influence on the structural characteristic parameters and mechanical properties of the scaffold. The modeling method of the multilevel Voronoi-lattice scaffold based on the Voronoi unit cell was proposed in this study to design the porous scaffold and meet the requirements of human bone morphology, mechanical properties, and actual manufacturing by adjusting factors V, F₁, and F₂. The proposed method can provide a feasible strategy for designing implants with suitable and similar morphologies and mechanical properties to cancellous bone.

Porous Structure of Cylindrical Particle Compacts

The porous compacts of non-spherical particles are frequently used in energy storage devices and other advanced applications. In the present work, the microstructures of compacts of monodisperse cylindrical particles are investigated. The cylindrical particles with various aspect ratios are generated using superquadrics, and the discrete element method was adopted to simulate the compacts formed under gravity deposition of randomly oriented particles. The Voronoi tessellation is then used to quantify the porous microstructure of compacts. With one exception, the median reduced free volume of Voronoi cells increases, and the median local packing density decreases for compacts composed of cylinders with a high aspect ratio, indicating a loose packing of long cylinders due to their mechanical interlocking during compaction. The obtained data are needed for further optimization of compact porous microstructure to improve the transport properties of compacts of non-spherical particles.

Cooperative success in epithelial public goods games

Cancer cells obtain mutations which rely on the production of diffusible growth factors to confer a fitness benefit. These mutations can be considered cooperative, and studied as public goods games within the framework of evolutionary game theory. The population structure, benefit function and update rule all influence the evolutionary success of cooperators. We model the evolution of cooperation in epithelial cells using the Voronoi tessellation model. Unlike traditional evolutionary graph theory, this allows us to implement global updating, for which birth and death events are spatially decoupled. We compare, for a sigmoid benefit function, the conditions for cooperation to be favoured and/or beneficial for well-mixed and structured populations. We find that when population structure is combined with global updating, cooperation is more successful than if there were local updating or the population were well-mixed. Interestingly, the qualitative behaviour for the well-mixed population and the Voronoi tessellation model is remarkably similar, but the latter case requires significantly lower incentives to ensure cooperation.

Simulation of Leather Visco-Elastic Behavior Based on Collagen Fiber-Bundle Properties and a Meso-Structure Network Model

Simulation-based prediction of mechanical properties is highly desirable for optimal choice and treatment of leather. Nowadays, this is state-of-the-art for many man-made materials. For the natural material leather, this task is however much more demanding due to the leather's high variability and its extremely intricate structure. Here, essential geometric features of the leather's meso-scale are derived from 3D images obtained by micro-computed tomography and subsumed in a parameterizable structural model. That is, the fiber-bundle structure is modeled. The structure model is combined with bundle properties derived from tensile tests. Then the effective leather visco-elastic properties are simulated numerically in the finite element representation of the bundle structure model with sliding contacts between bundles. The simulation results are validated experimentally for two animal types, several tanning procedures, and varying sample positions within the hide. Finally, a complete workflow for assessing leather quality by multi-scale simulation of elastic and visco-elastic properties is established and validated.

Optimized Atomic Partial Charges and Radii Defined by Radical Voronoi Tessellation of Bulk Phase Simulations

We present a novel method for the computation of well-defined optimized atomic partial charges and radii from the total electron density. Our method is based on a two-step radical Voronoi tessellation of the (possibly periodic) system and subsequent integration of the total electron density within each Voronoi cell. First, the total electron density is partitioned into the contributions of each molecule, and subsequently the electron density within each molecule is assigned to the individual atoms using a second set of atomic radii for the radical Voronoi tessellation. The radii are optimized on-the-fly to minimize the fluctuation (variance) of molecular and atomic charges. Therefore, our method is completely free of empirical parameters. As a by-product, two sets of optimized atomic radii are produced in each run, which take into account many specific properties of the system investigated. The application of an on-the-fly interpolation scheme reduces discretization noise in the Voronoi integration. The approach is particularly well suited for the calculation of partial charges in periodic bulk phase systems. We apply the method to five exemplary liquid phase simulations and show how the optimized charges can help to understand the interactions in the systems. Well-known effects such as reduced ion charges below unity in ionic liquid systems are correctly predicted without any tuning, empiricism, or rescaling. We show that the basis set dependence of our method is very small. Only the total electron density is evaluated, and thus, the approach can be combined with any electronic structure method that provides volumetric total electron densities-it is not limited to Hartree-Fock or density functional theory (DFT). We have implemented the method into our open-source software tool TRAVIS.

Numerical Investigation of Effective Thermal Conductivity of Strut-Based Cellular Structures Designed by Spatial Voronoi Tessellation

Porous materials possess light weight and excellent thermal insulation performance. For disordered porous structures, the number of seed points is an important design parameter which is closely related to the morphology and mean pore size of the structure. Based on the arrangement of points in three-dimensional space, seven kinds of structures were designed by spatial Voronoi tessellation in this paper. The effect of the number of seed points on effective thermal conductivity for Voronoi was studied. Numerical simulation was conducted to research the effects of structural porosity, filling material and structural orientation on the effective thermal conductivity and heat transfer characteristics. The results showed that the effective thermal conductivity is closely related to the porosity and the matrix material. Different number and arrangement of seed points make the structure have different anisotropic performance due to different thermal paths. In addition, required the least number of seed points was obtained for the designation of isotropic random Voronoi.

Evolution of cooperation in an epithelium

Cooperation is prevalent in nature, not only in the context of social interactions within the animal kingdom but also on the cellular level. In cancer, for example, tumour cells can cooperate by producing growth factors. The evolution of cooperation has traditionally been studied for well-mixed populations under the framework of evolutionary game theory, and more recently for structured populations using evolutionary graph theory (EGT). The population structures arising due to cellular arrangement in tissues, however, are dynamic and thus cannot be accurately represented by either of these frameworks. In this work, we compare the conditions for cooperative success in an epithelium modelled using EGT, to those in a mechanical model of an epithelium—the Voronoi tessellation (VT) model. Crucially, in this latter model, cells are able to move, and birth and death are not spatially coupled. We calculate fixation probabilities in the VT model through simulation and an approximate analytic technique and show that this leads to stronger promotion of cooperation in comparison with the EGT model.

Contact Area-Based Structural Analysis of Proteins and Their Complexes Using CAD-Score

Quantifying discrepancies between computationally derived and native (reference) structures is an essential step in the development and comparison of protein modeling and protein-protein docking methods. Measuring conformational differences of proteins or protein complexes is also important in other areas of structural biology such as molecular dynamics and crystallography. There are multiple scores to do that. However, nearly all of them, whether superposition-based (e.g., RMSD) or superposition-free, use distances to measure similarity. CAD-score is conceptually different as it uses physical contacts represented as contact areas. Such representation makes it possible to quantify differences of both structures and surfaces (e.g., protein-protein interfaces and binding sites) using the same framework. A number of studies have found CAD-score to be among the most robust scores. The method is implemented both as a web server and as standalone software available at http://bioinformatics.lt/software/cad-score . Here, we describe how to use the standalone CAD-score software for comparison and analysis of protein structures, interfaces, and binding sites.

Enhancing oncolytic virotherapy: Observations from a Voronoi Cell-Based model

Oncolytic virotherapy is a promising cancer treatment using genetically modified viruses. Unfortunately, virus particles rapidly decay inside the body, significantly hindering their efficacy. In this article, treatment perturbations that could overcome obstacles to oncolytic virotherapy are investigated through the development of a Voronoi Cell-Based model (VCBM). The VCBM derived captures the interaction between an oncolytic virus and cancer cells in a 2-dimensional setting by using an agent-based model, where cell edges are designated by a Voronoi tessellation. Here, we investigate the sensitivity of treatment efficacy to the configuration of the treatment injections for different tumour shapes: circular, rectangular and irregular. The model predicts that multiple off-centre injections improve treatment efficacy irrespective of tumour shape. Additionally, we investigate delaying the infection of cancer cells by modifying viral particles with a substance such as alginate (a hydrogel polymer used in a range of cancer treatments). Simulations of the VCBM show that delaying the infection of cancer cells, and thus allowing more time for virus dissemination, can improve the efficacy of oncolytic virotherapy. The simulated treatment noticeably decreases the tumour size with no increase in toxicity. Improving oncolytic virotherapy in this way allows for a more effective treatment without changing its fundamental essence.

Validation and Investigation on the Mechanical Behavior of Concrete Using a Novel 3D Mesoscale Method

The mechanical performance of concrete is strongly influenced by the geometry and properties of its components (namely aggregate, mortar, and Interfacial Transitional Zone (ITZ)) from the mesoscale viewpoint, and analyzing the material at that level should be a powerful tool for understanding macroscopic behavior. In this paper, a simple and highly efficient method is proposed for constructing realistic mesostructures of concrete. A shrinking process based on 3D Voronoi tessellation was employed to generate aggregates with random polyhedron and grading size, and reversely, an extending procedure was applied for ITZ generation. 3D mesoscale numerical simulation was conducted under a quasi-static load using an implicit solver which demonstrated the good robustness and feasibility of the presented model. The simulated results resembled favorably the corresponding experiments both in stress–strain curves and failure modes. Damage evolution analysis showed that the ITZ phase has profound influence on the damage behavior of concrete as damage initially develops from here and propagates to mortar. In addition, it was found that tensile damage is the principal factor of mortar failure while compressive damage is the principal factor of ITZ failure under compression.

A New Technique Based on Voronoi Tessellation to Assess the Space-Dependence of Categorical Variables

Based on a sample of geolocated elements, each of them labeled with a (not necessarily ordered) categorical feature, several indexes for assessing the relationship between the geolocation variables (latitude and longitude) and the categorical variable are evaluated. Among these indexes, a new one based on a Voronoi tessellation presents several advantages since it does not require a variable transformation or a previous discretization; in addition, simulations show that this index is considerably robust when compared with the previously known ones. Finally, the use of the presented indexes is also illustrated by analyzing the geolocation of communities in some communication networks derived from Call Detail Records.

Symmetry and Shannon Measure of Ordering: Paradoxes of Voronoi Tessellation

The Voronoi entropy for random patterns and patterns demonstrating various elements of symmetry was calculated. The symmetric patterns were characterized by the values of the Voronoi entropy being very close to those inherent to random ones. This contradicts the idea that the Voronoi entropy quantifies the ordering of the seed points constituting the pattern. Extension of the Shannon-like formula embracing symmetric patterns is suggested. Analysis of Voronoi diagrams enables the elements of symmetry of the patterns to be revealed.

Influence of Porosity on the Mechanical Behavior during Uniaxial Compressive Testing on Voronoi-Based Open-Cell Aluminium Foam

We have studied an application of the Voronoi tessellation method in the modeling of open-cell aluminium foam under uniaxial compressive loading. The Voronoi code was merged with computer-aided design (CAD) for converting the polyhedral model into an irregular open-cell cellular structure to create porous samples for compression testing simulations. Numerical simulations of the uniaxial compression uniformly over the upper surface of the sample in the z-axis direction at a constant 20 N load was realised. Samples with three different porosities (30%, 60% and 80%) were studied. A nonlinear elasto-plastic material model with perfect plasticity, without hardening, based on the von Mises yield criterion was applied below 10% strain. Corresponding stress–strain curves were observed and the influence of porosity on deformation mechanism was discussed. Samples with higher porosity exhibited significantly higher normal stress under the same load, and increased stress plateaus. An increase of porosity produced an increase of both compressive and tensile stresses and struts exhibited complex stress fields. Voronoi-based modeling was in accordance with experimental results in the literature in the case of the quasi-static condition and linear elastic region (below 1% strain). Further study is necessary to enable the simulation of real dynamic behaviour under all deformation regimes by using the Voronoi tessellation method.

LAGUERRE-INTERSECTION METHOD FOR IMPLICIT SOLVATION

Explicit solvent molecular dynamics simulations of a macromolecule are slow as the number of solvent atoms considered typically increases by order of magnitude. Implicit methods introduce surface-dependent corrections to the force field, gaining speed at the expense of accuracy. Properties such as molecular interface surfaces, volumes and cavities are captured by Laguerre tessellations of macromolecules. However, Laguerre cells of exterior atoms tend to be overly large or unbounded. Our method, the inclusion-exclusion based Laguerre-Intersection method, caps cells in a physically accurate manner by considering the intersection of the space-filling diagram with the Laguerre tessellation. We optimize an adjustable parameter, the weight, to ensure the areas and volumes of capped cells exposed to solvent are as close as possible, on average, to those computed from equilibrated explicit solvent simulations. The contact planes are radical planes, meaning that as the solvent weight is varied, interior cells remain constant. We test the consistency of our model using a high-quality trajectory of HIV-protease, a dimer with flexible loops and open-close transitions. We also compare our results with interval-arithmetic Gauss-Bonnet based method. Optimal solvent parameters quickly converge, which we use to illustrate the increased fidelity of the Laguerre-Intersection method over two recently proposed methods as compared to the explicit model.

Mechanics of epithelial tissue formation

A key process in the life of any multicellular organism is its development from a single egg into a full grown adult. The first step in this process often consists of forming a tissue layer out of randomly placed cells on the surface of the egg. We present a model for generating such a tissue, based on mechanical interactions between the cells, and find that the resulting cellular pattern corresponds to the Voronoi tessellation of the nuclei of the cells. Experimentally, we obtain the same result in both fruit flies and flour beetles, with a distribution of cell shapes that matches that of the model, without any adjustable parameters. Finally, we show that this pattern is broken when the cells grow at different rates.

Functional Data Approximation on Bounded Domains using Polygonal Finite Elements

We construct and analyze piecewise approximations of functional data on arbitrary 2D bounded domains using generalized barycentric finite elements, and particularly quadratic serendipity elements for planar polygons. We compare approximation qualities (precision/convergence) of these partition-of-unity finite elements through numerical experiments, using Wachspress coordinates, natural neighbor coordinates, Poisson coordinates, mean value coordinates, and quadratic serendipity bases over polygonal meshes on the domain. For a convex n-sided polygon, the quadratic serendipity elements have 2n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, rather than the usual n(n + 1)/2 basis functions to achieve quadratic convergence. Two greedy algorithms are proposed to generate Voronoi meshes for adaptive functional/scattered data approximations. Experimental results show space/accuracy advantages for these quadratic serendipity finite elements on polygonal domains versus traditional finite elements over simplicial meshes. Polygonal meshes and parameter coefficients of the quadratic serendipity finite elements obtained by our greedy algorithms can be further refined using an L₂-optimization to improve the piecewise functional approximation. We conduct several experiments to demonstrate the efficacy of our algorithm for modeling features/discontinuities in functional data/image approximation.

Enhancing Received Signal Strength-Based Localization through Coverage Hole Detection and Recovery

In wireless sensor networks (WSNs), Radio Signal Strength Indicator (RSSI)-based localization techniques have been widely used in various applications, such as intrusion detection, battlefield surveillance, and animal monitoring. One fundamental performance measure in those applications is the sensing coverage of WSNs. Insufficient coverage will significantly reduce the effectiveness of the applications. However, most existing studies on coverage assume that the sensing range of a sensor node is a disk, and the disk coverage model is too simplistic for many localization techniques. Moreover, there are some localization techniques of WSNs whose coverage model is non-disk, such as RSSI-based localization techniques. In this paper, we focus on detecting and recovering coverage holes of WSNs to enhance RSSI-based localization techniques whose coverage model is an ellipse. We propose an algorithm inspired by Voronoi tessellation and Delaunay triangulation to detect and recover coverage holes. Simulation results show that our algorithm can recover all holes and can reach any set coverage rate, up to 100% coverage.

Optimal and stable random pattern formations

We develop a graph model to describe the vast range of patterns observed in biological structures. For any given number of spotty patterns, a finite number of structures (optimal graphs) is precisely described. The construction of the optimal graphs is based on the minimization of the diffusion dissipation energy. The notion of geometrical stability of structures is introduced. It is demonstrated that the hexagonal array is stable and the square array is not. This explains the reason why the hexagonal array appears more frequently in practice than the square one.

Disease Risk Assessment Using a Voronoi-Based Network Analysis of Genes and Variants Scores

Much effort has been devoted to assess disease risk based on large-scale protein-protein network and genotype-phenotype associations. However, the challenge of risk prediction for complex diseases remains unaddressed. Here, we propose a framework to quantify the risk based on a Voronoi tessellation network analysis, taking into account the disease association scores of both genes and variants. By integrating ClinVar, SNPnexus, and DISEASES databases, we introduce a gene-variant map that is based on the pairwise disease-associated gene-variant scores. This map is clustered using Voronoi tessellation and network analysis with a threshold obtained from fitting the background Voronoi cell density distribution. We define the relative risk of disease that is inferred from the scores of the data points within the related clusters on the gene-variant map. We identify autoimmune-associated clusters that may interact at the system-level. The proposed framework can be used to determine the clusters that are specific to a subtype or contribute to multiple subtypes of complex diseases.

Limits of Applicability of the Voronoi Tessellation Determined by Centers of Cell Nuclei to Epithelium Morphology

It is well accepted that cells in the tissue can be regarded as tiles tessellating space. A number of approaches were developed to find an appropriate mathematical description of such cell tiling. A particularly useful approach is the so called Voronoi tessellation, built from centers of mass of the cell nuclei (CMVT), which is commonly used for estimating the morphology of cells in epithelial tissues. However, a study providing a statistically sound analysis of this method's accuracy is not available in the literature. We addressed this issue here by comparing a number of morphological measures of the cells, including area, perimeter, and elongation obtained from such a tessellation with identical measures extracted from direct imaging acquired by staining the cell membranes. After analyzing the shapes of 15,000 MDCK II epithelial cells under several conditions, we find that CMVT reasonably well reproduces many of the morphological properties of the tissue with an error that is between 10 and 15%. Moreover, cross-correlations between different morphological measures are reproduced qualitatively correctly by this method. However, all of the properties including the cell perimeters, number of neighbors, and anisotropy measures often suffer from systematic or size dependent errors. These discrepancies originate from the polygonal nature of the tessellation which sets the limits of the applicability of CMVT.

Spatial Heterogeneity Analysis in Evaluation of Cell Viability and Apoptosis for Colorectal Cancer Cells

In evaluation of cell viability and apoptosis, spatial heterogeneity is quantified for cancerous cells cultured in 3-D in vitro cell-based assays under the impact of anti-cancer agents. In 48-h experiments using human colorectal cancer cell lines of HCT-116, SW-620, and SW-480, incubated cells are divided into control and drug administered groups, to be grown in matrigel and FOLFOX solution, respectively. Our 3-D cell tracking and data acquisition system guiding an inverted microscope with a digital camera is utilized to capture bright field and fluorescent images of colorectal cancer cells at multiple time points. Identifying the locations of live and dead cells in captured images, spatial point process and Voronoi tessellation methods are applied to extract morphological features of in vitro cell-based assays. For the former method, spatial heterogeneity is quantified with the second-order functions of Poisson point process, whereas the deviation in the area of Voronoi polygons is computed for the latter. With both techniques, the results indicate that the spatial heterogeneity of live cell locations increases as the viability of in in vitro cell cultures decreases. On the other hand, a decrease is observed for the heterogeneity of dead cell locations with the decrease in cell viability. This relationship between morphological features of in vitro cell-based assays and cell viability can be used for drug efficacy measurements and utilized as a biomarker for 3-D in vitro microenvironment assays.

Homogeneity and EPR metrics for assessment of regular grids used in CW EPR powder simulations

CW EPR powder spectra may be approximated numerically using a spherical grid and a Voronoi tessellation-based cubature. For a given spin system, the quality of simulated EPR spectra depends on the grid type, size, and orientation in the molecular frame. In previous work, the grids used in CW EPR powder simulations have been compared mainly from geometric perspective. However, some grids with similar homogeneity degree generate different quality simulated spectra. This paper evaluates the grids from EPR perspective, by defining two metrics depending on the spin system characteristics and the grid Voronoi tessellation. The first metric determines if the grid points are EPR-centred in their Voronoi cells, based on the resonance magnetic field variations inside these cells. The second metric verifies if the adjacent Voronoi cells of the tessellation are EPR-overlapping, by computing the common range of their resonance magnetic field intervals. Beside a series of well known regular grids, the paper investigates a modified ZCW grid and a Fibonacci spherical code, which are new in the context of EPR simulations. For the investigated grids, the EPR metrics bring more information than the homogeneity quantities and are better related to the grids' EPR behaviour, for different spin system symmetries. The metrics' efficiency and limits are finally verified for grids generated from the initial ones, by using the original or magnetic field-constraint variants of the Spherical Centroidal Voronoi Tessellation method.