Analysis of fractal dimension of O2A glial cells differentiating in vitro

Fractal dimension is a quantitative measure of morphological complexity. Glial cells of the oligodendrocyte-type 2 astrocyte (O2A) lineage exhibit increasing morphological complexity as they differentiate in vitro. Enriched populations of O2A progenitor cells isolated from neonatal rat cerebral hemispheres or optic nerves were allowed to differentiate in vitro, and their fractal dimensions were measured over time. The fractal dimensions of the maturing cells correlated with perceived complexity; cells with elaborate process branching had larger fractal dimensions than cells with a simpler morphology. An analysis of changes in fractal dimension revealed distinct rates of growth for both oligodendrocytes and type 2 astrocytes. The fractal dimension remained constant over a 10-fold range in optical magnification, demonstrating that cultured O2A glial cells exhibit self-similarity, a defining characteristic of fractal objects. These results illustrate that fractal dimension analysis of maturing cell populations is a useful method for quantitatively describing the process of cell differentiation.     

Correcting for finite spatial scales of self-similarity when calculating fractal dimensions of real-world structures

Fractal geometry is a potentially valuable tool for quantitatively characterizing complex structures. The fractal dimension (D) can be used as a simple, single index for summarizing properties of real and abstract structures in space and time. Applications in the fields of biology and ecology range from neurobiology to plant architecture, landscape structure, taxonomy and species diversity. However, methods to estimate the D have often been applied in an uncritical manner, violating assumptions about the nature of fractal structures. The most common error involves ignoring the fact that ideal, i.e. infinitely nested, fractal structures exhibit self-similarity over any range of scales. Unlike ideal fractals, real-world structures exhibit self-similarity only over a finite range of scales.Here we present a new technique for quantitatively determining the scales over which real-world structures show statistical self-similarity. The new technique uses a combination of curve-fitting and tests of curvilinearity of residuals to identify the largest range of contiguous scales that exhibit statistical self-similarity. Consequently, we estimate D only over the statistically identified region of self-similarity and introduce the finite scale- corrected dimension (FSCD). We demonstrate the use of this method in two steps. First, using mathematical fractal curves with known but variable spatial scales of self-similarity (achieved by varying the iteration level used for creating the curves), we demonstrate that our method can reliably quantify the spatial scales of self-similarity. This technique therefore allows accurate empirical quantification of theoretical Ds. Secondly, we apply the technique to digital images of the rhizome systems of goldenrod (Solidago altissima). The technique significantly reduced variations in estimated fractal dimensions arising from variations in the method of preparing digital images. Overall, the revised method has the potential to significantly improve repeatability and reliability for deriving fractal dimensions of real-world branching structures.     

Development,persistence and regeneration of foraging ectomycorrhizal mycelial systems in soil microcosms

Development of extraradical mycelia of two strains each of Paxillus involutus and Suillus bovinus in ectomycorrhizal association with Pinus sylvestris seedlings was studied in two dimensions in non-sterile soil microcosms. There were significant inter- and intra-specific differences in extraradical mycelial growth and morphology. The mycelial systems of both strains of P. involutus were diffuse and extended more rapidly than those of S. bovinus. Depending on the strain, P. involutus mycelia were either highly plane filled, with high mass fractal dimension (a measure of space filling) or sparse, low mass fractal dimension systems. Older mycelial systems persisted as linear cords interlinking ectomycorrhizal tips. S. bovinus produced either a mycelium with a mixture of mycelial cords and diffuse fans that rapidly filled explorable area, or a predominately corded mycelium of minimal area cover. In the soil microcosms, mass fractal dimension and mycelial cover tended to increase with time, mycelia encountering litter having significantly greater values. Results are discussed in terms of the ecology of these fungi, their foraging activities and functional importance in forest ecosystems.     

Fractal properties of forest spatial structure

The definition of fractal dimension of natural objects, which enables to deal with scale dependence of fractal dimension is discussed. Abrupt changes of fractal dimension of spatial structure of terrestrial ecosystems are considered in the context of hierarchical paradigm. On this ground the procedure is proposed for segmentation of a territory, which takes into account the scale dependence of spatial variability of ecological parameters. Using remotely sensed data — normalized difference vegetation index (NDVI) and thermal radiation in the infrared band — fractal dimensions and critical scales are evaluated for different forest types with the help of software, developed for this purpose. The results obtained corroborate the potentialities of fractal approach in ecology. These methods and results can be used for discrimination of remotely sensed data; but further investigations, including detailed comparison of fractal characteristics of remotely sensed forest images with results of on-site field studies are necessary to validate them.     

Simulation of growth of a filamentous fungus in 3 dimensions

The tridimensional growth of a filamentous fungus was simulated, based on a model for the evolution of the microscopic morphology of Trichoderma reesei. When supplemented with a spatial representation of growth, the model correctly simulates the evolution from a single spore to a pellet. Diffusion of oxygen is included in the model. The simulated tridimensional structures have a fractal nature; and the fractal dimension, determined by a box-counting method, increases during growth. The fractal dimension only depends on the mass of the pellet and is not affected by model parameters such as tip extension rate and branching frequency. Realistic pictures are obtained and the radius of the pellet increases at a constant rate. The influence of model parameters (tip extension rate, branching frequency, minimum porosity) on dissolved oxygen concentration profiles, biomass concentration profiles, rate at which the pellet diameter increases, and the evolution of the fractal dimension was determined. The dissolved oxygen profiles were found to be very different from the profiles, obtained by assuming a homogenous biomass distribution within the pellet. Finally, the formation of pellets from spore aggregates is calculated and the size of the spore aggregate is found to only influence the time needed before the appearance of a pellet and not its morphology. (c) 1997 John Wiley & Sons, Inc.     

A single-cell model to study changes in neuronal fractal dimension

Many methods have been developed to quantify neuronal morphology: measurement of neurite length, neurite number, etc. However, none of these approaches provides a comprehensive view of the complexity of neuronal morphology. In this work we have analyzed the evaluation of fractal dimension (D) as a tool to represent and quantify changes in complexity of the dendritic arbor, in in vitro cultures grown under low-density conditions. Neurons grown in isolation developed a bipolar morphology corresponding to a fractal dimension close to the unit. The analysis showed that neuronal complexity increased when cells were incubated with a depolarizing potassium concentration and there was a correlation with an increase in fractal dimension (D5 mM KCl = 1.08 +/- 0.01, D25 mM KCl =1.25 +/- 0.01). We conclude that fractal dimension is a suitable parameter to quantify changes in neuronal morphological complexity.     

Morphological measurements on Penicillium chrysogenum broths by rheology and filtration methods

The morphology parameters of mycelial culture (Penicillium chrysogenum) were measured and quantified by rheology and filtration methods. Two of the morphology parameters obtained from rheology measurements, delta defined by the Casson equation and delta* defined by intrinsic viscosity, were found to vary systematically with broth age and with the observed morphology by microscopy. Three of the filtration parameters, hyphal density, Kozeny constant, and index of compressibility, are demonstrated as sensitive indicators of the broth age and mycelial morphology. Two of the morphology parameters, delta and delta*, were used to cross-correlate with hyphal density. Because various mycelial fermentations require different growth morphologies (pellet and filament) for optimum product yield and the morphology of mycelial broths varies with broth age, it is suggested that these morphology parameters could be used to represent the morphology of mycelial broths quantitatively. (c) 1993 John Wiley & Sons, Inc.     

Quantifying the branching frequency of virtual filamentous microbes using fractal analysis

The productivity of an industrial fermentation process involving a filamentous microbe is heavily dependent on the morphological form adopted by the organism. The development of systems capable of rapidly and accurately characterizing morphology within a given process represents a significant challenge, as the complex phenotypes that are manifested are not easily quantified. Conventional parameters employed in these analyses are often of limited value, as they reveal little about the branching behavior of the organism; an important consideration given the demonstrated link between branching frequency and metabolite production. In this study, the influence of branching behavior on the spatial distribution of mycelia grown in silico is examined through fractal analysis. It is demonstrated that fractal dimension, quantified based on the frequency distribution of parameterized boundary curves, and lacunarity act as robust estimators of branching behavior. The analysis can, in theory, be applied to any morphological form, providing universally applicable process parameters for more complete data acquisition. Biotechnol. Bioeng. 2013; 110: 437–447. © 2012 Wiley Periodicals, Inc.     

Fractal Organization of Cultivated in vitro Spiculogenous Cells and Larval Spicules of the Sea Urchin Strongylocentrotus nudus

The morphological patterns of the cultivated cells of primary mesenchyme and the spicules of the larval skeleton of the sea urchin Strongylocentrotus nudus were quantified, and the value of their fractal dimensions (D) was determined with ImageJ 1.20s software. It was shown that during cytodifferentiation, the values of D in the fractal (fractional) dimension, which reflects the complex spatial organization of the spiculogenous mesenchyme elements in two-dimensional space, increase to values close to 1.7. The invertible treatment with cytochalasin, which destroys the system of the actin filaments, suppresses the normal control of biomineralization and causes a complex form of spicules, the fractal dimension of which varies within 1.5–1.6. Thus, the determination of the fractal dimension value serves as evidence of the fractional essence of the patterns studied, quantifies the spatially complex organization of cells and their assemblies during morphogenesis, and allows us to estimate the variation in the spicule morphology after cytochalasin treatment.     

Brownian Dynamics Computer Simulations of Quenched Lennard-Jones-like Fluids: I Morphology and Local Structural Evolution

Abstract

The structural characteristics during phase separation of a model colloidal system were investigated using Brownian dynamics simulation. The structures that formed were analysed using the radial distribution function and structure factor in separate time periods after the quench. The data were interpreted in terms of scale-invariancy and density inhomogeneities. The systems, which consisted of a gas-like phase and dense liquid or solid-like regions, developed with a highly interconnected morphology during the simulations. The aggregate morphology was sensitive to the range of the attractive part of the potential and the position in the phase diagram after the quench. The long-range 12:6 potential induced compact structures with thick filaments, whereas the systems generated using the shorter-ranged 24:12 and 36:18 potentials persisted in a more diffuse network and also evolved more slowly with time. The fractal dimensions were quite high, typically close to 3. The 24:12 and 36:18 potential systems developed regions of local crystalline order which formed contemporaneously with the more global morphological changes. In contrast, at low temperatures the particles of the longer-range 12:6 potential became trapped in glass-like states during the course of the morphological changes in the system. The value of the characteristic lengthscale with time exponent, α, was found to be dependent on the temperature, density and interaction potential and therefore cannot be described as ‘universal’.     

Fungal fractal morphology of pellet formation in Aspergillus niger

Mycelial fractal values were compared to the conventional fungal morphological parameters: average total mycelial length, average number of tips and average growth unit. The fractal values were between 1.47 to 1.3 for the various submerged culture conditions of Aspergillus niger. The average pellet diameter was 1.4 mm at the fractal value of 1.47. The mycelia with fractal values close to 1 were less branched and slim.     

Three-dimensional biofilm structure quantification

Quantitative parameters describing biofilm physical structure have been extracted from three-dimensional confocal laser scanning microscopy images and used to compare biofilm structures, monitor biofilm development, and quantify environmental factors affecting biofilm structure. Researchers have previously used biovolume, volume to surface ratio, roughness coefficient, and mean and maximum thicknesses to compare biofilm structures. The selection of these parameters is dependent on the availability of software to perform calculations. We believe it is necessary to develop more comprehensive parameters to describe heterogeneous biofilm morphology in three dimensions. This research presents parameters describing three-dimensional biofilm heterogeneity, size, and morphology of biomass calculated from confocal laser scanning microscopy images. This study extends previous work which extracted quantitative parameters regarding morphological features from two-dimensional biofilm images to three-dimensional biofilm images. We describe two types of parameters: (1) textural parameters showing microscale heterogeneity of biofilms and (2) volumetric parameters describing size and morphology of biomass. The three-dimensional features presented are average (ADD) and maximum diffusion distances (MDD), fractal dimension, average run lengths (in X, Y and Z directions), aspect ratio, textural entropy, energy and homogeneity. We discuss the meaning of each parameter and present the calculations in detail. The developed algorithms, including automatic thresholding, are implemented in software as MATLAB programs which will be available at site prior to publication of the paper.     

樟子松人工林树冠结构的分形分析

基于樟子松人工林7块固定标准地中的31株解析木的树冠体积和叶量,以幂函数关系(F=Av^(D/3))建立了预估树冠表面积的分形维数。同时根据生物量实测数据,建立预估叶量的生物模型Lw=0.180397D^3045903H^-1.67348。基于枝解析、树干解析数据,动态地预估了一年、二年、三年前的树冠体积,并结合树冠体积、叶量的这种幂函数关系可以动态地预估一年、二年、三年前树冠表面积的分形维数,从而反映出树冠结构的动态变化规律。为了了解不同分级样木的分维数变化情况,利用2003年调查的4块生物量标准地数据,根据单株树木各个枝条占据的空间体积与该枝条的带叶枝干重的关系,计算了各标准地不同分级样木树冠的分维数。为探讨单株样木树冠的分维数的计算提供了一种可行方法。树冠的分维数作为表征树冠的动态生长变化是一有用和可靠的指标。     

利用IKONOS卫星数据和分形方法研究南麂岛土地覆盖状况

研究南麂岛的土地覆盖类型及其空间分布和结构特征。利用具有1米空间分辨率的IKONOS卫星遥感数据,提取南麂岛的植被覆盖和土地利用信息,获得草地、灌木林地、庄稼地和居民地等主要土地覆盖类型及其分布图。然后利用分形几何方法建立南麂岛土地覆盖类型特性分析模型,从斑块的面积效应、覆盖类型的分形分析、单个斑块的分形分析和覆盖类型分形特征差异显著性等方面进行分析讨论。研究结果表明,南麂岛的草地和灌木林地的分形维数较大,而庄稼地和居民地的分形维数较小,说明草地和灌木林地的斑块的结构特征和边界比庄稼地和居民地更为复杂。进一步研究表明,斑块的分形特性与其受人类活动的干扰程度密切相关。     

凉山半细毛羊初生重性状分形特征分析

首次以非线性理论中的分形理论对凉山半细毛羊的初生重数据进行分析, 计算了1996~2004年间初生重性状的信息维数、关联维数、无标度区及性状测度范围。结果表明: (1) 从1996~2004年凉山半细毛羊的初生重性状的信息维数都集中在0.66529~0.90675, 而且无标度区较大, 测度范围广。该结果说明, 每年羔羊初生重信息维数都较大, 群体存在十分丰富的变异, 变异范围广, 具有十分巨大的育种潜力; (2) 相应的关联维数则集中在0.62438~0.86528之间, 表明群体内个体遗传结构具有较强的相关性; (3) 这两个分形维数能够分别从两个不同的角度揭示群体遗传结构的分形特征。     

不同光照条件下假俭草生长格局的分形研究

应用分形几何的原理和方法,研究不同光照条件下假俭草的生长格局。结果表明,假俭草具有自相似性的生长过程。不同光照条件下,假俭草具有不同的分形维数和相应的生物量积累规律。生物量积累速率随着分形维数的增大而增大。分形维数同时反映了假俭草的分枝能力和对空间的占据能力。假俭草种群具有依据外界条件而调节生长格局的生态适应能力。     

Microbial growth patterns described by fractal geometry.

Fractal geometry has made important contributions to understanding the growth of inorganic systems in such processes as aggregation, cluster formation, and dendritic growth. In biology, fractal geometry was previously applied to describe, for instance, the branching system in the lung airways and the backbone structure of proteins as well as their surface irregularity. This investigation applies the fractal concept to the growth patterns of two microbial species, Streptomyces griseus and Ashbya gossypii. It is a first example showing fractal aggregates in biological systems, with a cell as the smallest aggregating unit and the colony as an aggregate. We find that the global structure of sufficiently branched mycelia can be described by a fractal dimension, D, which increases during growth up to 1.5. D is therefore a new growth parameter. Two different box-counting methods (one applied to the whole mass of the mycelium and the other applied to the surface of the system) enable us to evaluate fractal dimensions for the aggregates in this analysis in the region of D = 1.3 to 2. Comparison of both box-counting methods shows that the mycelial structure changes during growth from a mass fractal to a surface fractal.     

The Fractal Patterns of Words in a Text: A Method for Automatic Keyword Extraction

A text can be considered as a one dimensional array of words. The locations of each word type in this array form a fractal pattern with certain fractal dimension. We observe that important words responsible for conveying the meaning of a text have dimensions considerably different from one, while the fractal dimensions of unimportant words are close to one. We introduce an index quantifying the importance of the words in a given text using their fractal dimensions and then ranking them according to their importance. This index measures the difference between the fractal pattern of a word in the original text relative to a shuffled version. Because the shuffled text is meaningless (i.e., words have no importance), the difference between the original and shuffled text can be used to ascertain degree of fractality. The degree of fractality may be used for automatic keyword detection. Words with the degree of fractality higher than a threshold value are assumed to be the retrieved keywords of the text. We measure the efficiency of our method for keywords extraction, making a comparison between our proposed method and two other well-known methods of automatic keyword extraction.     

间作对桑树和谷子生长和光合日变化的影响

以桑树和谷子为研究材料,探讨了大田条件下,桑树-谷子间作对桑树和谷子的干物质生产、土地利用率和光合日变化的影响.结果表明：桑树-谷子间作条件下,间作桑树的株高、地茎、根长和枝条数分别比单作桑树增加了6.0%、13.7%、6.8%和14.8%,且间作桑树的产叶量比单作桑树增加了31.3%;间作谷子与单作比较,其株高和根长的变化不大.桑树-谷子间作增加了土地当量比,提高了土地利用率.单作、间作桑树和谷子叶片在12:00时均表现出明显的光合午休现象,且单作桑树的光合午休现象比间作桑树严重.桑树-谷子间作提高了中午时桑树叶片气孔导度和水分利用率,增加了桑树光合碳同化能力,抑制了桑树叶片实际光化学效率、电子传递速率和最大光化学效率的下降,从而减缓了桑树的“光合午休”现象.桑树 谷子间作能明显提高桑树叶片的光合生产能力.