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.     

Root Systems and Fractals: How Reliable are Calculations of Fractal Dimensions?

Fractal dimensions of the root systems of Betula populifoliaand Betula alleghaniensis grown at two levels of soil nutrientswere calculated from tracings of root systems (Fitter and Stickland,1992a) and from roots removed from the soil using a pin-board(Tatsumi, Yamauchi and Kono, 1989). The fractal dimensions obtainedfrom these two methods showed similar qualitative patterns betweenspecies and nutrient treatments, but were not significantlycorrelated with one another. These observations suggest thatcurrent methods used for estimating the fractal dimensions ofplant root systems are not yet reliable. It is suggested thatthe calculation of fractal dimensions needs to be carried outin three dimensions in order to appropriately quantify the fractaldimensions of plant root systems.Copyright 1994, 1999 AcademicPress Betula populifolia Marsh (gray birch), Betula alleghaniensis Britton (yellow birch), fractal dimension, root system     

Fractal analysis of a voltage-dependent potassium channel from cultured mouse hippocampal neurons.

The kinetics of ion channels have been widely modeled as a Markov process. In these models it is assumed that the channel protein has a small number of discrete conformational states and the kinetic rate constants connecting these states are constant. In the alternative fractal model the spontaneous fluctuations of the channel protein at many different time scales are represented by a kinetic rate constant k = At1-D, where A is the kinetic setpoint and D the fractal dimension. Single-channel currents were recorded at 146 mM external K+ from an inwardly rectifying, 120 pS, K+ selective, voltage-sensitive channel in cultured mouse hippocampal neurons. The kinetics of these channels were found to be statistically self-similar at different time scales as predicted by the fractal model. The fractal dimensions were approximately 2 for the closed times and approximately 1 for the open times and did not depend on voltage. For both the open and closed times the logarithm of the kinetic setpoint was found to be proportional to the applied voltage, which indicates that the gating of this channel involves the net inward movement of approximately one negative charge when this channel opens. Thus, the open and closed times and the voltage dependence of the gating of this channel are well described by the fractal model.     

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。     

A fractal model for the characterization of mycelial morphology

A new technique based on a fractal model has been developed for the quantification of the macroscopic morophology of mycelia. The morphological structuring is treated as a fractal object, and the fractal dimension, determined by an ultrasonic scattering procedure developed for the purpose, serves as a quantitative morphological index. Experimental observations reported earlier and simulations of mycelial growth, carried out using a probabilistic-geometric growth model developed for the purpose, both validate the applicability of the fractal model. In experiments with three different species, the fractal dimensions of pelletous structures were found to be in the range 1.45-2.0 and those of filamentous structures were in the range 1.9-2.7, with values around 2.0 representing mixed morphologies. Fractal dimensions calculated from simulated mycelia are in rough agreement with these ranges. The fractal dimension is also found to be relatively insensitive to the biomass concentration, as seen by dilution of the original broths. The relation between morphology and filtration properties of the broths has also been studied. The fractal dimension shows a strong correlation with the index of cake compressibility and with the Kozeny constant, two filtration parameters that are known to be morphology dependent. This technique could thus be used to develop correlations between the morphology, represented by the fractal dimension, and important morphology-dependent process variables. (c) 1993 John Wiley & Sons, Inc.     

Self similarity of protein surfaces.

Scaling properties of the surfaces of 53 proteins are studied on the basis of experimentally determined 3-D structures of these biological macromolecules. It is found that the surfaces show self similarity within a yardstick range of 1.5 A less than epsilon less than 15 A. The self similarity is measured by the fractal dimension D of the surface. Two different algorithms for the determination of the fractal dimension are applied, both based on cubic yardstick particles. One is related to the contact surface (CS), which was first introduced by Connolly (17), while the other corresponds to the solvent accessible surface (SAS) of Richards (9). The fractal dimensions of both are different. While the CS type approach leads to relatively high values of D in the range 2.5 to 2.6 the SAS approach gives fractal dimensions of D approximately 2.     

Mechanical characterization of network formation during heat-induced gelation of whey protein dispersions

The formation of gel network structures during isothermal heating of whey protein aqueous dispersions was probed by mechanical spectroscopy. It was anticipated that the pathway of the sol-to-gel transition of whey protein dispersions is quite different from that of ordinary cross-linking polymers (e.g., percolation-type transition), since aqueous solutions of native whey proteins have been shown to be highly structured even before gelation, in our previous study. At 20 degrees C, aqueous dispersions of beta-lactoglobulin, the major whey protein, and those of whey protein isolate (WPI), a mixture of whey proteins, exhibited solid-like mechanical spectra, i.e., the predominant storage modulus G' over the loss modulus G", in a certain range of the frequency omega (1-100 rad/s), regardless of the presence or absence of added NaCl. The existence of the added salt was, however, a critical factor for determining transitions in mechanical spectra during gelation at 70 degrees C. beta-Lactoglobulin dispersions in 0.1 mol/dm(3) NaCl maintained the solid-like nature during the entire gelation process and, after passing through the gelation point, satisfied parallel power laws (G' approximately G" approximately omega(n)) that have been proposed for a critical gel (i.e., the gel at the gelation point) that possesses a self-similar or fractal network structure. In contrast, beta-lactoglobulin dispersions without added salt exhibited a transition from solid-like [G'(omega) > G"(omega)] to liquid-like [G'(omega) < G"(omega)] mechanical spectra before gelation, but no parallel power law behavior was recognized at the gelation point. During extended heating time (aging), beta-lactoglobulin gels with 0.1 mol/dm(3) NaCl showed deviations from the parallel power laws, while spectra of gels without added NaCl approached the parallel power laws, suggesting that post-gelation reactions also significantly affect gel network structures. A percolation-type sol-to-gel transition was found only for WPI dispersions without added salt.     

分子进化研究：Ⅱ.5SrRNA序列的分形与分子进化的关系

本文根据分形理论的原理和方法,在对现行的计算核酸序列分维的方法进行修改的基础上,对各类生物的80余种5SrRNA序列的分维进行了计算,并结合耗散结构理论就其分维与分子进化的关系问题进行了研究和探讨。作者认为,5SrRNA序列的分维与其分子进化间的关系是一种复杂的非线性关系,在分子进化的过程中,序列的分维表现为随机涨落。     

Passive ionic properties of frog retinal pigment epithelium.

The isolated pigment epithelium and choroid of frog was mounted in a chamber so that the apical surfaces of the epithelial cells and the choroid were exposed to separate solutions. The apical membrane of these cells was penetrated with microelectrodes and the mean apical membrane potential was --88 mV. The basal membrane potential was depolarized by the amount of the transepithelial potential (8--20 mV). Changes in apical and basal cell membrane voltage were produced by changing ion concentrations on one or both sides of the tissue. Although these voltage changes were altered by shunting and changes in membrane resistance, it was possible to estimate apical and basal cell membrane and shunt resistance, and the relative ionic conductance Ti of each membrane. For the apical membrane: TK approximately equal to 0.52, THCO3 approximately equal to 0.39 and TNa approximately equal to 0.05, and its specific resistance was estimated to be 6000--7000 omega cm2. For the basal membrane: TK approximately equal to 0.90 and its specific resistance was estimated to be 400--1200 omega cm2. From the basal potassium voltage responses the intracellular potassium concentration was estimated at 110 mM. The shunt resistance consisted of two pathways: a paracellular one, due to the junctional complexes and another, around the edge of the tissue, due to the imperfect nature of the mechanical seal. In well-sealed tissues, the specific resistance of the shunt was about ten times the apical plus basal membrane specific resistances. This epithelium, therefore, should be considered "tight". The shunt pathway did not distinguish between anions (HCO--3, Cl--, methylsulfate, isethionate) but did distinguish between Na+ and K+.     

Self-similar colony morphogenesis by gram-negative rods as the experimental model of fractal growth by a cell population.

The ability to form a fractal colony was shown to be common among several species of the family Enterobacteriaceae. Bacterial spreading growth in a two-dimensional field of nutrient concentration was indicated to be important for this experimental self-similar morphogenesis. As a basic analogy, the diffusion-limited aggregation model was suggested. Fractal dimensions of colonies were mostly in the range of values from 1.7 to 1.8, similar to those of the two-dimensional diffusion-limited aggregation model. Bacterial characteristics and culture conditions inducing changes in fractal patterns and growth rates were identified. The contribution of the bacterial multicellular nature to fractal morphogenesis is discussed.     

Self-similar colony morphogenesis by gram-negative rods as the experimental model of fractal growth by a cell population.

The ability to form a fractal colony was shown to be common among several species of the family Enterobacteriaceae. Bacterial spreading growth in a two-dimensional field of nutrient concentration was indicated to be important for this experimental self-similar morphogenesis. As a basic analogy, the diffusion-limited aggregation model was suggested. Fractal dimensions of colonies were mostly in the range of values from 1.7 to 1.8, similar to those of the two-dimensional diffusion-limited aggregation model. Bacterial characteristics and culture conditions inducing changes in fractal patterns and growth rates were identified. The contribution of the bacterial multicellular nature to fractal morphogenesis is discussed.     

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.     

Parametric resonance and amplification of periodic disturbances in membranes containing ion channels with inactivation

Parametric resonance and amplification of periodic perturbations in the membrane transport of ions through channels with inactivation was studied in computational experiments. It has been shown that a periodic change in the membrane capacitance or in the applied electric current with a frequency approximately 2 omega 0 (omega 0--the own angular frequency of the membrane) may excite stable self-oscillations in the membrane with a frequency of approximately omega 0. For this to occur, the degree of the capacitance modulation m or the amplitude of the applied current i0 must exceed some critical values mcr and i0cr. Excitation of self-oscillations by alternating electric current of the frequency approximately 2 omega 0 has the characteristics of parametric resonance. This can be explained by the fact that the equivalent membrane inductance depends on ionic current and displays periodic changes with a frequency approximately 2 omega 0, as also does the current. Small-amplitude periodic changes in the capacitance (m less than mcr) with frequencies approximately 2 omega 0 may result in significant amplification of periodic perturbations with frequencies approximately omega 0.     

Explicitly Accounting for Pixel Dimension in Calculating Classical and Fractal Landscape Shape Metrics

Different summarized shape indices, like mean shape index (MSI) and area weighted mean shape index (AWMSI) can change over multiple size scales. This variation is important to describe scale heterogeneity of landscapes, but the exact mathematical form of the dependence is rarely known. In this paper, the use of fractal geometry (by the perimeter and area Hausdorff dimensions) made us able to describe the scale dependence of these indices. Moreover, we showed how fractal dimensions can be deducted from existing MSI and AWMSI data. In this way, the equality of a multiscale tabulated MSI and AWMSI dataset and two scale-invariant fractal dimensions has been demonstrated.     

Estimation of the junctional resistance between electrically coupled receptor cells in Necturus taste buds

Junctional resistance between coupled receptor cells in Necturus taste buds was estimated by modeling the results from single patch pipette voltage clamp studies on lingual slices. The membrane capacitance and input resistance of coupled taste receptor cells were measured to monitor electrical coupling and the results compared with those calculated by a simple model of electrically coupled taste cells. Coupled receptor cells were modeled by two identical receptor cells connected via a junctional resistance. On average, the junctional resistance was approximately 200-300 M omega. This was consistent with the electrophysiological recordings. A junctional resistance of 200-300 M omega is close to the threshold for Lucifer yellow dye-coupling detection (approximately 500 M omega). Therefore, the true extent of coupling in taste buds might be somewhat greater than that predicted from Lucifer yellow dye coupling. Due to the high input resistance of single taste receptor cells (> 1 G omega), a junctional resistance of 200-300 M omega assures a substantial electrical communication between coupled taste cells, suggesting that the electrical activity of coupled cells might be synchronized.     

云贵鹅耳枥种群分布格局的分形特征

应用分形理论中的计盒维数和信息维数探讨了贵阳喀斯特山地贵鹅耳枥种群分布格局的分形特征。结果表明,贵鹅耳枥种群的分布格局具有分形特征,其计盒维数为1．1853－1．7419,信息维数为1．1961－1．7051。集群型的贵鹅耳枥种群的计盒维数和信息维数均比随机型的高。计盒维数定量地反映了贵鹅耳枥种群占据生态空间的能力,信息维数则揭示了该种群格局强度的尺度变化程度和表征了种群个体分布的非均匀性。这两种维数方法都适用于贵鹅耳枥种群分布格局分形特征的定量描述。     

The catalytic site of rat hepatic lauric acid omega-hydroxylase. Protein versus prosthetic heme alkylation in the omega-hydroxylation of acetylenic fatty acids

Cytochrome P-450LA omega purified from clofibrate-induced rat liver oxidizes lauric acid to 11- and 12-hydroxydodecanoic acid in approximately a 1:17 ratio at a rate of 20 nmol/nmol P-450/min. In contrast, cytochrome P-450b oxidizes lauric acid much more slowly (0.5 nmol/nmol P-450/min) to an 8:1 mixture of the same metabolites. Western blot analysis indicates that P-450LA omega accounts for 1-2 and 16-30%, respectively, of the total cytochrome P-450 in uninduced and clofibrate-induced rat liver. Cytochrome b5 increases the efficiency of omega-hydroxylation but not the rate of catalytic turnover. Incubation of the enzyme with 10-undecynoic acid (10-UDYA) results in loss of approximately 45% of the enzymatic activity but none of the enzyme chromophore. Approximately 1 mol of 1,11-undecandioic acid is produced per mole of inactivated enzyme. This extraordinary inactivation efficiency is confirmed by NADPH consumption studies. Approximately 0.5 equivalents of label are covalently bound to the enzyme when it is incubated with 14C-labeled 10-UDYA. 11-Dodecenoic acid appears not to be a substrate for cytochrome P-450LA omega but is oxidized, presumably by a contaminating isozyme, to a 10:1 mixture of 11,12-epoxydodecanoic acid and 12-oxododecanoic acid. The results suggest the presence of two closely related P-450LA omega enzymes, only one of which is susceptible to inactivation by 10-UDYA. They also indicate that cytochrome P-450LA omega has a highly structured active site that sterically suppresses omega-1-hydroxylation in order to deliver the oxygen to the thermodynamically disfavored terminal carbon. Protein rather than heme alkylation follows from this reaction regiospecificity.     

北京地区植被景观斑块形状的分形分析

 借助GIS软件ARC／INFO、利用分形理论对北京地区的植被景观进行了分析,并对各种类型不同尺度上的斑块复杂性以及可能的尺度域进行了评价。结果表明：计算分维的两种方法得到的结果具有很强的正线性相关性,分维与平均斑块面积也呈显著的正相关;并且大多数相近类型(即同一大类内的各类型)斑块的分维也相近。本研究还识别出两个尺度域,一个是斑块面积小于2.7km2(大约),另一个是斑块面积大于2.7km2(大约)。后一个域中的斑块明显比前一个域中的斑块复杂。出现这种现象的原因主要是由于大斑块经常出现不同类型的斑块相互嵌套,而小斑块则很少出现甚至不出现这种现象。     

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.