Self-similar mitochondrial DNA

We show that repeated sequences, like palindromes (local repetitions) and homologies between two different nucleotide sequences (motifs along the genome), compose a self-similar (fractal) pattern in mitochondrial DNA. This self-similarity comes from the looplike structures distributed along the genome. The looplike structures generate scaling laws in a pseudorandom DNA walk constructed from the sequence, called a Lévy flight. We measure the scaling laws from the generalized fractal dimension and singularity spectrum for mitochondrial DNA walks for 35 different species. In particular, we report characteristic loop distributions for mammal mitochondrial genomes.     

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。     

保安湖莲群丛分布格局分形特征的初步研究

应用非线性科学中的分形几何理论,以保安湖扁担塘湖汊中的莲群丛为对象,研究该群丛小尺度水平格局的分形特征。主要应用计盒维数和信息维数公式计算群丛中莲种群和菱种群的个体分布格局的分维值,莲的计盒维数为1.92,信息维数为1.88;菱的计盒维数为1.04,信息维数为1.11.表明在采样区莲的空间占有程度远大于菱,是该时期的优势种,而菱则成为伴生种。莲在各尺度上的分布较均匀。最后讨论了莲的分维值在连续样方上的变化。       

Perfusion heterogeneity in rat lungs assessed from the distribution of 4-microm-diameter latex particles.

Pulmonary vascular perfusion has been shown to follow a fractal distribution down to a resolution of 0.5 cm(3) (5E11 microm(3)). We wanted to know whether this distribution continued down to tissue volumes equivalent to that of an alveolus (2E5 microm(3)). To investigate this, we used confocal microscopy to analyze the spatial distribution of 4-microm-diameter fluorescent latex particles trapped within rat lung microvessels. Particle distributions were analyzed in tissue volumes that ranged from 1.7E2 to 2.8E8 microm(3). The analysis resulted in fractal plots that consisted of two slopes. The left slope, encompassing tissue volumes less than 7E5 microm(3), had a fractal dimension of 1.50 +/- 0.03 (random distribution). The right slope, encompassing tissue volumes greater than 7E5 microm(3), had a fractal dimension of 1.29 +/- 0.04 (nonrandom distribution). The break point at 7E5 microm(3) corresponds closely to a tissue volume equivalent to that of one alveolus. We conclude that perfusion distribution is random at tissue volumes less than that of an alveolus and nonrandom at tissue volumes greater than that of an alveolus.     

Improving Accuracy and Precision in Estimating Fractal Dimension of Animal movement paths

It is difficult to watch wild animals while they move, so often biologists analyse characteristics of animal movement paths. One common path characteristic used is tortuousity, measured using the fractal dimension (D). The typical method for estimating fractal D, the divider method, is biased and imprecise. The bias occurs because the path length is truncated. I present a method for minimising the truncation error. The imprecision occurs because sometimes the divider steps land inside the bends of curves, and sometimes they miss the curves. I present three methods for minimising this variation and test the methods with simulated correlated random walks. The traditional divider method significantly overestimates fractal D when paths are short and the range of spatial scales is narrow. The best method to overcome these problems consists of walking the dividers forwards and backwards along the path, and then estimating the path length remaining at the end of the last divider step.     

Visual discrimination of fractal borders.

The ideas of fractals and fractal dimension are here translated into the realm of visual psychophysics. Borders between two fields of different luminance were used. Because of the finite grain of the visual system, fractal dimension need be defined only within a certain size range. For a fractal dimension of 1.15, the just-detectable difference in fractal dimension was found to be about 0.0085, rising to about 0.015 for a fractal dimension of 1.25. Reducing exposure duration from 1 s to 0.33 s decreases sensitivity to differences in fractal dimension, but there was no gain in increasing the exposure duration. Good visual observers who are naive to the task require some training before reaching optimal performance. The ability to discriminate fractal dimension differs between fractal edges of the same fractal dimension that were generated with differing statistical programs. Even after considerable training, an observer makes 29% errors when asked to distinguish a fractal edge generated with a Gaussian random walk from one with a rectangular random walk. Gaussian random walk fractals can be more easily distinguished from Poissonian and Cauchy ones.     

Determination of the fractal dimension of membrane protein aggregates using fluorescence energy transfer.

It is demonstrated that fluorescence resonance energy transfer may be used to determine the fractal dimension of aggregates of membrane-bound proteins. Theoretical and experimental results are presented for two different experimental designs: energy transfer between proteins and energy transfer from lipids to proteins. For energy transfer between proteins the lattice spacing must be known independently for a fractal dimension to be uniquely determined, and this represents a disadvantage to this experimental design. Results are presented for the calcium ATPase and a fractal dimension of 1.9 is estimated for ATPase aggregates by assuming a lattice spacing of 50 A. Energy transfer from lipids to protein provides a means of estimating the length of the "coast-line" of the aggregate. In this case the fractal dimension is uniquely determined from a log-log plot. An analysis of data for bacteriohodopsin reconstituted in phospholipid vesicles gives a fractal dimension of 1.6. The structural basis of the value for the fractal dimension is discussed for these two systems. These techniques provide a means of assessing the nature of protein-protein interactions in membranous systems.     

Local fractal dimension based ECG arrhythmia classification

We propose a local fractal dimension based nearest neighbor classifier for ECG based classification of arrhythmia. Local fractal dimension (LFD) at each sample point of the ECG waveform is taken as the feature. A nearest neighbor algorithm in the feature space is used to find the class of the test ECG beat. The nearest neighbor is found based on the RR-interval-information-biased Euclidean distance, proposed in the current work. Based on the two algorithms used for estimating the LFD, two classification algorithms are validated in the current work, viz. variance based fractal dimension estimation based nearest neighbor classifier and power spectral density based fractal dimension estimation based nearest neighbor classifier. Their performances are evaluated based on various figures of merit. MIT-BIH (Massachusetts Institute of Technology - Boston’s Beth Israel Hospital) Arrhythmia dataset has been used to validate the algorithms. Along with showing good performance against all the figures of merit, the proposed algorithms also proved to be patient independent in the sense that the performance is good even when the test ECG signal is from a patient whose ECG is not present in the training ECG dataset.     

Fractals and the analysis of growth paths

A simple practical method exists for classifying and comparing planar curves composed of connected line segments. This method assigns, a single numberD, the fractal dimension, to each curve.D=log(n)/[log(n)+log(d/L)], where:n is the number of line segments,L is the total length of the line segments, andd is the planar diameter of the curve (the greatest distance between any two endpoints). At one end of the spectrum, for straight line curves,D=1; at the other end of the spectrum, for random walk curves,D→2. Standard statistics are done on the logarithms of the fractal dimension [log(D)]. With this measure, trails of biological movement, such as the growth paths of the cells and the paths of wandering organisms, can be analyzed to determine the likelihood that these trails are random walks and also to compare the straightness of the trails before and after experimental interventions.     

保安湖—湖湾大型水生植物群落格局的研究

用地质统计学中的半方差和双对数半方差图对保安湖-湖湾(黄风口)的大型水生植物群落格局进行了研究,在不同的尺度上对群落及其主要组成种类的空间异质性进行了定量的描述。结果表明:保安湖黄风口大型水生植物群落的双对数半方差图存在线性区域,群落具有分形特征(自相似性).在不同的尺度下,群落和各组成种类空间格局具有不同的分形维数值,异质性程度存在差异。但其异质性程度不高,建议进行群落调查时,样方尺度应取520m;单一种群调查时的样方尺度取380m.       

Evaluation of self-similar features in time series of serum growth hormone and prolactin levels by fractal analysis: Effect of delayed sleep and complexity of diurnal variation

We assayed the diurnal concentrations of growth hormone (GH) and prolactin (PRL) in 6 healthy male volunteers to evaluate the self-similar features in the time series of each hormone on the basis of fractal theory and to determine the fractal dimension as an index of the complexity of the diurnal variation. In addition, we assessed the effects of a 6-hour delay in the sleep period on the complexity of the diurnal variaton of these hormones. There was a statistically significant fractal feature in the serum levels of GH both under the nocturnal-sleep and delayed-sleep conditions in all subjects. The time series of the serum PRL concentrations also showed a statistically significant fractal feature under the nocturnal-sleep and delayed-sleep conditions in all subjects. The fractal dimensions of the patterns of the GH or PRL levels were 1.879 and 1.929 or 1.754 and 1.785 under the nocturnal-sleep and delayed-sleep conditions, respectively. Two-way ANOVA revealed no significant difference in the fractal dimension between the two sleep conditions but did reveal a significant difference between the fractal dimensions of the GH and PRL levels. These results showed (1) that delayed sleep had no significant effect on the complexity of the diurnal pattern of these hormones, and (2) that the diurnal pattern of the GH levels was more complex than that of the PRL levels.     

Fractal analysis of the uterine contractions.

The fractal dimension D may be calculated in many ways, since its strict definition, the Hausdorff definition is too complicated for practical estimation. In this paper we perform a comparative study often methods of fractal analysis of time series. In Benoit, a commercial program for fractal analysis, five methods of computing fractal dimension of time series (rescaled range analysis, power spectral analysis, roughness-length, variogram methods and wavelet method) are available. We have implemented some other algorithms for calculating D: Higuchi's fractal dimension, relative dispersion analysis, running fractal dimension, method based on mathematical morphology and method based on intensity differences. For biomedical signals results obtained by means of different algorithms are different, but consistent.     

Biological factors underlying regularity and chaos in aquatic ecosystems: Simple models of complex dynamics

This work is focused on the processes underlying the dynamics of spatially inhomogeneous plankton communities. We demonstrate that reaction-diffusion mathematical models are an appropriate tool for searching and understanding basic mechanisms of complex spatio-temporal plankton dynamics and fractal properties of planktivorous fish school walks.     

Fractal geometry of root systems: Field observations of contrasting genotypes of common bean (Phaseolus vulgaris L.) grown under different phosphorus regimes

Root growth and architecture are important for phosphorus acquisition due to the relative immobility of P in the soil. Fractal geometry is a potential new approach to the analysis of root architecture. Substantial genetic variation in root growth and architecture has been observed in common bean. Common bean (Phaseolus vulgaris L.) genotypes with contrasting root architecture were grown under moderate and low P conditions in a field experiment. Linear and planar fractal dimension were measured by tracing root intercepts with vertical planes. Linear fractal dimension increased over time in efficient genotypes, but remained fairly constant over time in inefficient genotypes. Planar fractal dimension increased over time for all genotypes, but was higher in efficient than inefficient genotypes at the end of the experiment. Planar fractal dimension of medium P plants was found to correlate with shoot P content indicating fractal dimension to be a possible indicator for root P uptake. The increasing fractal dimension over time indicates that fractal analysis is a sensitive measure of root branching intensity. A less destructive method for acquisition of data that allows for continuous analysis of fractal geometry and thereby screening for more P efficient genotypes in the field is suggested. This method will allow the researcher to conduct fractal analysis and still complete field trials with final yield evaluation.     

Fractal dimension in human cerebellum measured by magnetic resonance imaging

Fractal dimension has been used to quantify the structures of a wide range of objects in biology and medicine. We measured fractal dimension of human cerebellum (CB) in magnetic resonance images of 24 healthy young subjects (12 men and 12 women). CB images were resampled to a series of image sets with different 3D resolutions. At each resolution, the skeleton of the CB white matter was obtained and the number of pixels belonging to the skeleton was determined. Fractal dimension of the CB skeleton was calculated using the box-counting method. The results indicated that the CB skeleton is a highly fractal structure, with a fractal dimension of 2.57 +/- 0.01. No significant difference in the CB fractal dimension was observed between men and women.     

基于RS和GIS的开封市土地覆盖分形

以RS和GIS为技术手段,利用开封市0.61m空间分辨率的QuickBird卫星遥感数据,采用分形几何方法研究了斑块的面积效应和覆盖类型分形的关系,并对覆盖类型的分形特征差异等进行了分析。结果表明,开封市防护林地和农田的平均斑块面积较大,分维数也较大,而水体的分维数较小,说明防护林和农田斑块的边界结构特征比水体更为复杂。斑块的分维值具有尺度依赖性,同一类型中大的斑块往往具有较大的分维值,其原因是大斑块经常出现不同类型的斑块相互嵌套,小斑块则很少出现甚至不出现这种现象。     

The Quasi-Fractal Structure of Fish Brain Neurons

The box-counting method for calculating the fractal dimension (D) with the ImageJ 1.20s software is used as a tool for quantitative analysis of the neuronal morphology in the fish brain. The fractal dimension was determined for several types of neurons in the brain of two teleost species, Pholidapus dybowskii and Oncorhynchus keta. These results were compared with those obtained for some neurons of the human brain. The fractal (fractional) dimension (D), as a quantitative index of filling of two-dimensional space by the black and white image of a cell, is shown to vary from 1.22 to 1.72 depending on the type of neuron. The fractal dimension reaches its maximum in less specialized neurons that carry out a number of different functions. On the other hand, highly specialized neurons display a relatively low fractal dimension. Thus, the fractal dimension serves as a numerical measure of the spatial complexity of the neuron and correlates with the morphofunctional organization of the cell.     

Modelling applicability of fractal analysis to efficiency of soil exploration by roots

BACKGROUND AND AIMS: Fractal analysis allows calculation of fractal dimension, fractal abundance and lacunarity. Fractal analysis of plant roots has revealed correlations of fractal dimension with age, topology or genotypic variation, while fractal abundance has been associated with root length. Lacunarity is associated with heterogeneity of distribution, and has yet to be utilized in analysis of roots. In this study, fractal analysis was applied to the study of root architecture and acquisition of diffusion-limited nutrients. The hypothesis that soil depletion and root competition are more closely correlated with a combination of fractal parameters than by any one alone was tested. MODEL: The geometric simulation model SimRoot was used to dynamically model roots of various architectures growing for up to 16 d in three soil types with contrasting nutrient mobility. Fractal parameters were calculated for whole roots, projections of roots and vertical slices of roots taken at 0, 2.5 and 5 cm from the root origin. Nutrient depletion volumes, competition volumes, and relative competition were regressed against fractal parameters and root length. KEY RESULTS: Root length was correlated with depletion volume, competition volume and relative competition at all times. In analysis of three-dimensional, projected roots and 0 cm slices, log(fractal abundance) was highly correlated with log(depletion volume) when times were pooled. Other than this, multiple regression yielded better correlations than regression with single fractal parameters. Correlations decreased with age of roots and distance of vertical slices from the root origin. Field data were also examined to see if fractal dimension, fractal abundance and lacunarity can be used to distinguish common bean genotypes in field situations. There were significant differences in fractal dimension and fractal abundance, but not in lacunarity. CONCLUSIONS: These results suggest that applying fractal analysis to research of soil exploration by root systems should include fractal abundance, and possibly lacunarity, along with fractal dimension.     

Ontogenetic changes in the fractal geometry of the bronchial tree in Rattus norvegicus

Respiration and metabolism change dramatically over the course of the development of vertebrates. In mammals these changes may be ascribed to organogenesis and differentiation of structures involved in gas exchange and transport and the increase in size. Since young as well as mature individuals must be well-designed if the species is to survive, the physiological changes during the development should be matched with geometrical or structural adjustments of the respiratory system. The aim of this study was to evaluate changes in the fractal geometry of the bronchial tree during the postnatal development of the rat. The average fractal dimension of the bronchial tree of the rats was 1.587, but that of juveniles was larger than that of the adults. We found a significant negative correlation between age and fractal dimension. This correlation could be considered be misleading because of the difficulty of separating age/body size effects. Nevertheless, because fractal dimensions of the bronchial tree of rabbits and humans are known to be similar, 1.58 and 1.57 respectively, the body size effect may be nil. To our knowledge, this is the first report of ontogenetic changes in the fractal dimension of the bronchial tree in mammals.