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1.  Evaluation of the Dispersional Analysis Method for Fractal Time Series 
Annals of biomedical engineering  1995;23(4):491-505.
Fractal signals can be characterized by their fractal dimension plus some measure of their variance at a given level of resolution. The Hurst exponent, H, is <0.5 for rough anticorrelated series, >0.5 for positively correlated series, and =0.5 for random, white noise series. Several methods are available: dispersional analysis, Hurst rescaled range analysis, autocorrelation measures, and power special analysis. Short data sets are notoriously difficult to characterize; research to define the limitations of the various methods is incomplete. This numerical study of fractional Brownian noise focuses on determining the limitations of the dispersional analysis method, in particular, assessing the effects of signal length and of added noise on the estimate of the Hurst coefficient, H, (which ranges from 0 to 1 and is 2 − D, where D is the fractal dimension). There are three general conclusions: (i) pure fractal signals of length greater than 256 points give estimates of H that are biased but have standard deviations less than 0.1; (ii) the estimates of H tend to be biased toward H = 0.5 at both high H (>0.8) and low H (<0.5), and biases are greater for short time series than for long; and (iii) the addition of Gaussian noise (H = 0.5) degrades the signals: for those with negative correlation (H < 0.5) the degradation is great, the noise has only mild degrading effects on signals with H > 0.6, and the method is particularly robust for signals with high H and long series, where even 100% noise added has only a few percent effect on the estimate of H. Dispersional analysis can be regarded as a strong method for characterizing biological or natural time series, which generally show long-range positive correlation.
PMCID: PMC3756095  PMID: 7486356
Time series analysis; Autocovariance; Gaussian and fractional Brownian noise; Correlation; Hurst coefficient; Fractal dimension; Statistics
2.  The most frequent short sequences in non-coding DNA 
Nucleic Acids Research  2009;38(4):1172-1181.
The purpose of this work is to determine the most frequent short sequences in non-coding DNA. They may play a role in maintaining the structure and function of eukaryotic chromosomes. We present a simple method for the detection and analysis of such sequences in several genomes, including Arabidopsis thaliana, Caenorhabditis elegans, Drosophila melanogaster and Homo sapiens. We also study two chromosomes of man and mouse with a length similar to the whole genomes of the other species. We provide a list of the most common sequences of 9–14 bases in each genome. As expected, they are present in human Alu sequences. Our programs may also give a graph and a list of their position in the genome. Detection of clusters is also possible. In most cases, these sequences contain few alternating regions. Their intrinsic structure and their influence on nucleosome formation are not known. In particular, we have found new features of short sequences in C. elegans, which are distributed in heterogeneous clusters. They appear as punctuation marks in the chromosomes. Such clusters are not found in either A. thaliana or D. melanogaster. We discuss the possibility that they play a role in centromere function and homolog recognition in meiosis.
doi:10.1093/nar/gkp1094
PMCID: PMC2831315  PMID: 19966278
3.  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.
doi:10.1098/rspb.1997.0212
PMCID: PMC1688706
Branching Structures Fractal Dimension Self-Simularity Solidago Altissima Spatial Scales
4.  Surface characterization of proteins using multi-fractal property of heat-denatured aggregates 
Bioinformation  2008;2(9):379-383.
Multi-fractal property of heat-denatured protein aggregates (HDPA) is characteristic of its individual form. The visual similarity between digitally generated microscopic images of HDPA with that of surface-image of its individual X-ray structures in protein databank (PDB) displayed using Visual Molecular Dynamics (VMD) viewer is the basis of the study. We deigned experiments to view the fractal nature of proteins at different aggregate scales. Intensity based multi-fractal dimensions (ILMFD) extracted from various planes of digital microscopic images of protein aggregates were used to characterize HDPA into different classes. Moreover, the ILMFD parameters extracted from aggregates show similar classification pattern to digital images of protein surface displayed by VMD viewer using PDB entry. We discuss the use of irregular patterns of heat-denatured aggregate proteins to understand various surface properties in native proteins.
PMCID: PMC2533056  PMID: 18795110
protein structure; multi-fractal dimension; protein aggregate; digital image processing; light microscopy; degree of differentiation; graphical pair-wise class comparison
5.  Analyzing Self-Similar and Fractal Properties of the C. elegans Neural Network 
PLoS ONE  2012;7(10):e40483.
The brain is one of the most studied and highly complex systems in the biological world. While much research has concentrated on studying the brain directly, our focus is the structure of the brain itself: at its core an interconnected network of nodes (neurons). A better understanding of the structural connectivity of the brain should elucidate some of its functional properties. In this paper we analyze the connectome of the nematode Caenorhabditis elegans. Consisting of only 302 neurons, it is one of the better-understood neural networks. Using a Laplacian Matrix of the 279-neuron “giant component” of the network, we use an eigenvalue counting function to look for fractal-like self similarity. This matrix representation is also used to plot visualizations of the neural network in eigenfunction coordinates. Small-world properties of the system are examined, including average path length and clustering coefficient. We test for localization of eigenfunctions, using graph energy and spacial variance on these functions. To better understand results, all calculations are also performed on random networks, branching trees, and known fractals, as well as fractals which have been “rewired” to have small-world properties. We propose algorithms for generating Laplacian matrices of each of these graphs.
doi:10.1371/journal.pone.0040483
PMCID: PMC3465333  PMID: 23071485
6.  Comparison of C. elegans and C. briggsae Genome Sequences Reveals Extensive Conservation of Chromosome Organization and Synteny 
PLoS Biology  2007;5(7):e167.
To determine whether the distinctive features of Caenorhabditis elegans chromosomal organization are shared with the C. briggsae genome, we constructed a single nucleotide polymorphism–based genetic map to order and orient the whole genome shotgun assembly along the six C. briggsae chromosomes. Although these species are of the same genus, their most recent common ancestor existed 80–110 million years ago, and thus they are more evolutionarily distant than, for example, human and mouse. We found that, like C. elegans chromosomes, C. briggsae chromosomes exhibit high levels of recombination on the arms along with higher repeat density, a higher fraction of intronic sequence, and a lower fraction of exonic sequence compared with chromosome centers. Despite extensive intrachromosomal rearrangements, 1:1 orthologs tend to remain in the same region of the chromosome, and colinear blocks of orthologs tend to be longer in chromosome centers compared with arms. More strikingly, the two species show an almost complete conservation of synteny, with 1:1 orthologs present on a single chromosome in one species also found on a single chromosome in the other. The conservation of both chromosomal organization and synteny between these two distantly related species suggests roles for chromosome organization in the fitness of an organism that are only poorly understood presently.
Author Summary
The importance of chromosomal organization in the fitness of a species is only poorly understood. The publication of the C. elegans genome sequence in 1998 revealed features of higher level organization that suggested its chromosomes were organized into distinct domains. Chromosome arms were accumulating changes more rapidly than the centers of chromosomes. In this paper, we have compared the organization of the nematode C. briggsae genome with that of C. elegans. By building a genetic map based on DNA variations between two strains of C. briggsae, and by using that map to organize the draft genome sequence of C. briggsae published in 2003, we found the following: (1) Intrachromosomal rearrangements are frequent within and even between arms but are less common within central regions and between arms and centers. (2) Genes have remained overwhelmingly on the same chromosomes. (3) The distinctive features that distinguish C. elegans arms from centers also are seen in C. briggsae chromosomes. The conservation of these features between these two species, despite the approximately 100 million years since their most recent common ancestor, provides clear evidence of the selective advantages of the domain architecture of chromosomes. The continuing association of genes on the same chromosomes suggests that this may also be advantageous.
The conservation of both chromosomal organization and synteny between two distantly related species suggests roles for chromosome organization in the fitness of an organism.
doi:10.1371/journal.pbio.0050167
PMCID: PMC1914384  PMID: 17608563
7.  Fractal Physiology and the Fractional Calculus: A Perspective 
This paper presents a restricted overview of Fractal Physiology focusing on the complexity of the human body and the characterization of that complexity through fractal measures and their dynamics, with fractal dynamics being described by the fractional calculus. Not only are anatomical structures (Grizzi and Chiriva-Internati, 2005), such as the convoluted surface of the brain, the lining of the bowel, neural networks and placenta, fractal, but the output of dynamical physiologic networks are fractal as well (Bassingthwaighte et al., 1994). The time series for the inter-beat intervals of the heart, inter-breath intervals and inter-stride intervals have all been shown to be fractal and/or multifractal statistical phenomena. Consequently, the fractal dimension turns out to be a significantly better indicator of organismic functions in health and disease than the traditional average measures, such as heart rate, breathing rate, and stride rate. The observation that human physiology is primarily fractal was first made in the 1980s, based on the analysis of a limited number of datasets. We review some of these phenomena herein by applying an allometric aggregation approach to the processing of physiologic time series. This straight forward method establishes the scaling behavior of complex physiologic networks and some dynamic models capable of generating such scaling are reviewed. These models include simple and fractional random walks, which describe how the scaling of correlation functions and probability densities are related to time series data. Subsequently, it is suggested that a proper methodology for describing the dynamics of fractal time series may well be the fractional calculus, either through the fractional Langevin equation or the fractional diffusion equation. A fractional operator (derivative or integral) acting on a fractal function, yields another fractal function, allowing us to construct a fractional Langevin equation to describe the evolution of a fractal statistical process. Control of physiologic complexity is one of the goals of medicine, in particular, understanding and controlling physiological networks in order to ensure their proper operation. We emphasize the difference between homeostatic and allometric control mechanisms. Homeostatic control has a negative feedback character, which is both local and rapid. Allometric control, on the other hand, is a relatively new concept that takes into account long-time memory, correlations that are inverse power law in time, as well as long-range interactions in complex phenomena as manifest by inverse power-law distributions in the network variable. We hypothesize that allometric control maintains the fractal character of erratic physiologic time series to enhance the robustness of physiological networks. Moreover, allometric control can often be described using the fractional calculus to capture the dynamics of complex physiologic networks.
doi:10.3389/fphys.2010.00012
PMCID: PMC3059975  PMID: 21423355
fractals; fractional calculus; physiology; allometric control; power-law statistics
8.  Flocculation control study based on fractal theory*  
A study on flocculation control based on fractal theory was carried out. Optimization test of chemical coagulant dosage confirmed that the fractal dimension could reflect the flocculation degree and settling characteristics of aggregates and the good correlation with the turbidity of settled effluent. So that the fractal dimension can be used as the major parameter for flocculation system control and achieve self-acting adjustment of chemical coagulant dosage. The fractal dimension flocculation control system was used for further study carried out on the effects of various flocculation parameters, among which are the dependency relationship among aggregates fractal dimension, chemical coagulant dosage, and turbidity of settled effluent under the conditions of variable water quality and quantity. And basic experimental data were obtained for establishing the chemical coagulant dosage control model mainly based on aggregates fractal dimension.
doi:10.1631/jzus.2005.B1038
PMCID: PMC1390449  PMID: 16187420
Aggregates; Flocculation control; Fractal dimension; Image analysis; Turbidity
9.  The Genome Sequence of Caenorhabditis briggsae: A Platform for Comparative Genomics 
PLoS Biology  2003;1(2):e45.
The soil nematodes Caenorhabditis briggsae and Caenorhabditis elegans diverged from a common ancestor roughly 100 million years ago and yet are almost indistinguishable by eye. They have the same chromosome number and genome sizes, and they occupy the same ecological niche. To explore the basis for this striking conservation of structure and function, we have sequenced the C. briggsae genome to a high-quality draft stage and compared it to the finished C. elegans sequence. We predict approximately 19,500 protein-coding genes in the C. briggsae genome, roughly the same as in C. elegans. Of these, 12,200 have clear C. elegans orthologs, a further 6,500 have one or more clearly detectable C. elegans homologs, and approximately 800 C. briggsae genes have no detectable matches in C. elegans. Almost all of the noncoding RNAs (ncRNAs) known are shared between the two species. The two genomes exhibit extensive colinearity, and the rate of divergence appears to be higher in the chromosomal arms than in the centers. Operons, a distinctive feature of C. elegans, are highly conserved in C. briggsae, with the arrangement of genes being preserved in 96% of cases. The difference in size between the C. briggsae (estimated at approximately 104 Mbp) and C. elegans (100.3 Mbp) genomes is almost entirely due to repetitive sequence, which accounts for 22.4% of the C. briggsae genome in contrast to 16.5% of the C. elegans genome. Few, if any, repeat families are shared, suggesting that most were acquired after the two species diverged or are undergoing rapid evolution. Coclustering the C. elegans and C. briggsae proteins reveals 2,169 protein families of two or more members. Most of these are shared between the two species, but some appear to be expanding or contracting, and there seem to be as many as several hundred novel C. briggsae gene families. The C. briggsae draft sequence will greatly improve the annotation of the C. elegans genome. Based on similarity to C. briggsae, we found strong evidence for 1,300 new C. elegans genes. In addition, comparisons of the two genomes will help to understand the evolutionary forces that mold nematode genomes.
With the Caenorhabditis briggsae genome now in hand, C. elegans biologists have a powerful new research tool to refine their knowledge of gene function in C. elegans and to study the path of genome evolution
doi:10.1371/journal.pbio.0000045
PMCID: PMC261899  PMID: 14624247
10.  Protein Coding Sequence Identification by Simultaneously Characterizing the Periodic and Random Features of DNA Sequences 
Most codon indices used today are based on highly biased nonrandom usage of codons in coding regions. The background of a coding or noncoding DNA sequence, however, is fairly random, and can be characterized as a random fractal. When a gene-finding algorithm incorporates multiple sources of information about coding regions, it becomes more successful. It is thus highly desirable to develop new and efficient codon indices by simultaneously characterizing the fractal and periodic features of a DNA sequence. In this paper, we describe a novel way of achieving this goal. The efficiency of the new codon index is evaluated by studying all of the 16 yeast chromosomes. In particular, we show that the method automatically and correctly identifies which of the three reading frames is the one that contains a gene.
doi:10.1155/JBB.2005.139
PMCID: PMC1184046  PMID: 16046819
11.  Self-organization of developing embryo using scale-invariant approach 
Background
Self-organization is a fundamental feature of living organisms at all hierarchical levels from molecule to organ. It has also been documented in developing embryos.
Methods
In this study, a scale-invariant power law (SIPL) method has been used to study self-organization in developing embryos. The SIPL coefficient was calculated using a centro-axial skew symmetrical matrix (CSSM) generated by entering the components of the Cartesian coordinates; for each component, one CSSM was generated. A basic square matrix (BSM) was constructed and the determinant was calculated in order to estimate the SIPL coefficient. This was applied to developing C. elegans during early stages of embryogenesis. The power law property of the method was evaluated using the straight line and Koch curve and the results were consistent with fractal dimensions (fd). Diffusion-limited aggregation (DLA) was used to validate the SIPL method.
Results and conclusion
The fractal dimensions of both the straight line and Koch curve showed consistency with the SIPL coefficients, which indicated the power law behavior of the SIPL method. The results showed that the ABp sublineage had a higher SIPL coefficient than EMS, indicating that ABp is more organized than EMS. The fd determined using DLA was higher in ABp than in EMS and its value was consistent with type 1 cluster formation, while that in EMS was consistent with type 2.
doi:10.1186/1742-4682-8-17
PMCID: PMC3126770  PMID: 21635789
12.  Microarchitecture Parameters Describe Bone Structure and Its Strength Better Than BMD 
The Scientific World Journal  2012;2012:502781.
Introduction and Hypothesis. Some papers have shown that bone mineral density (BMD) may not be accurate in predicting fracture risk. Recently microarchitecture parameters have been reported to give information on bone characteristics. The aim of this study was to find out if the values of volume, fractal dimension, and bone mineral density are correlated with bone strength. Methods. Forty-two human bone samples harvested during total hip replacement surgery were cut to cylindrical samples. The geometrical mesh of layers of bone mass obtained from microCT investigation and the volumes of each layer and fractal dimension were calculated. The finite element method was applied to calculate the compression force F causing ε = 0.8% strain. Results. There were stronger correlations for microarchitecture parameters with strength than those for bone mineral density. The values of determination coefficient R2 for mean volume and force were 0.88 and 0.90 for mean fractal dimension and force, while for BMD and force the value was 0.53. The samples with bigger mean bone volume of layers and bigger mean fractal dimension of layers (more complex structure) presented higher strength. Conclusion. The volumetric and fractal dimension parameters better describe bone structure and strength than BMD.
doi:10.1100/2012/502781
PMCID: PMC3361288  PMID: 22654618
13.  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.
Images
PMCID: PMC195579  PMID: 1599243
14.  Representation of cloned genomic sequences in two sequencing vectors: correlation of DNA sequence and subclone distribution. 
Nucleic Acids Research  1997;25(15):2960-2966.
Representation of subcloned Caenorhabditis elegans and human DNA sequences in both M13 and pUC sequencing vectors was determined in the context of large scale genomic sequencing. In many cases, regions of subclone under-representation correlated with the occurrence of repeat sequences, and in some cases the under-representation was orientation specific. Factors which affected subclone representation included the nature and complexity of the repeat sequence, as well as the length of the repeat region. In some but not all cases, notable differences between the M13 and pUC subclone distributions existed. However, in all regions lacking one type of subclone (either M13 or pUC), an alternate subclone was identified in at least one orientation. This suggests that complementary use of M13 and pUC subclones would provide the most comprehensive subclone coverage of a given genomic sequence.
PMCID: PMC146865  PMID: 9224593
15.  Fractal parameters and vascular networks: facts & artifacts 
Background
Several fractal and non-fractal parameters have been considered for the quantitative assessment of the vascular architecture, using a variety of test specimens and of computational tools. The fractal parameters have the advantage of being scale invariant, i.e. to be independent of the magnification and resolution of the images to be investigated, making easier the comparison among different setups and experiments.
Results
The success of several commercial and/or free codes in computing the fractal parameters has been tested on well known exact models. Based on such a preliminary study, we selected the code Frac-lac in order to analyze images obtained by visualizing the angiogenetic process occurring in chick Chorio Allontoic Membranes (CAM), assumed to be paradigmatic of a realistic 2D vascular network. Among the parameters investigated, the fractal dimension Df proved to be the most robust estimator for CAM vascular networks. Moreover, only Df was able to discriminate between effective and elusive increases in vascularization after drug-induced angiogenic stimulations on CAMs.
Conclusion
The fractal dimension Df is likely to be the most promising tool for monitoring the effectiveness of anti-angiogenic therapies in various clinical contexts.
doi:10.1186/1742-4682-5-12
PMCID: PMC2490683  PMID: 18637183
16.  Fractal Dimension and Vessel Complexity in Patients with Cerebral Arteriovenous Malformations 
PLoS ONE  2012;7(7):e41148.
The fractal dimension (FD) can be used as a measure for morphological complexity in biological systems. The aim of this study was to test the usefulness of this quantitative parameter in the context of cerebral vascular complexity. Fractal analysis was applied on ten patients with cerebral arteriovenous malformations (AVM) and ten healthy controls. Maximum intensity projections from Time-of-Flight MRI scans were analyzed using different measurements of FD, the Box-counting dimension, the Minkowski dimension and generalized dimensions evaluated by means of multifractal analysis. The physiological significance of this parameter was investigated by comparing values of FD first, with the maximum slope of contrast media transit obtained from dynamic contrast-enhanced MRI data and second, with the nidus size obtained from X-ray angiography data. We found that for all methods, the Box-counting dimension, the Minkowski dimension and the generalized dimensions FD was significantly higher in the hemisphere with AVM compared to the hemisphere without AVM indicating that FD is a sensitive parameter to capture vascular complexity. Furthermore we found a high correlation between FD and the maximum slope of contrast media transit and between FD and the size of the central nidus pointing out the physiological relevance of FD. The proposed method may therefore serve as an additional objective parameter, which can be assessed automatically and might assist in the complex workup of AVMs.
doi:10.1371/journal.pone.0041148
PMCID: PMC3399805  PMID: 22815946
17.  Selective modulation of cell response on engineered fractal silicon substrates 
Scientific Reports  2013;3:1461.
A plethora of work has been dedicated to the analysis of cell behavior on substrates with ordered topographical features. However, the natural cell microenvironment is characterized by biomechanical cues organized over multiple scales. Here, randomly rough, self-affinefractal surfaces are generated out of silicon,where roughness Ra and fractal dimension Df are independently controlled. The proliferation rates, the formation of adhesion structures, and the morphology of 3T3 murine fibroblasts are monitored over six different substrates. The proliferation rate is maximized on surfaces with moderate roughness (Ra ~ 40 nm) and large fractal dimension (Df ~ 2.4); whereas adhesion structures are wider and more stable on substrates with higher roughness (Ra ~ 50 nm) and lower fractal dimension (Df ~ 2.2). Higher proliferation occurson substrates exhibiting densely packed and sharp peaks, whereas more regular ridges favor adhesion. These results suggest that randomly roughtopographies can selectively modulate cell behavior.
doi:10.1038/srep01461
PMCID: PMC3598004  PMID: 23492898
18.  Quantification of the fractal nature of mycelial aggregation in Aspergillus niger submerged cultures 
Background
Fractal geometry estimates have proven useful in studying the growth strategies of fungi in response to different environments on soil or on agar substrates, but their use in mycelia grown submerged is still rare. In the present study, the effects of certain important fermentation parameters, such as the spore inoculum level, phosphate and manganese concentrations in the medium, on mycelial morphology of the citric acid producer Aspergillus niger were determined by fractal geometry. The value of employing fractal geometry to describe mycelial structures was examined in comparison with information from other descriptors including classic morphological parameters derived from image analysis.
Results
Fractal analysis of distinct morphological forms produced by fermentation conditions that influence fungal morphology and acid production, showed that the two fractal dimensions DBS (box surface dimension) and DBM (box mass dimension) are very sensitive indexes, capable of describing morphological differences. The two box-counting methods applied (one applied to the whole mass of the mycelial particles and the other applied to their surface only) enabled evaluation of fractal dimensions for mycelial particles in this analysis in the region of DBS = 1.20–1.70 and DBM = 1.20–2.70. The global structure of sufficiently branched mycelia was described by a single fractal dimension D, which did not exceed 1.30. Such simple structures are true mass fractals (DBS = DBM = D) and they could be young mycelia or dispersed forms of growth produced by very dense spore inocula (108–109 spores/ml) or by addition of manganese in the medium. Mycelial clumps and pellets were effectively discriminated by fractal analysis. Fractal dimension values were plotted together with classic morphological parameters derived from image analysis for comparisons. Their sensitivity to treatment was analogous to the sensitivity of classic morphological parameters suggesting that they could be equally used as morphological descriptors.
Conclusion
Starting from a spore, the mycelium develops as a mass fractal and, depending on culture conditions, it either turns to a surface fractal or remains a mass fractal. Since fractal dimensions give a measure of the degree of complexity and the mass filling properties of an object, it may be possible that a large number of morphological parameters which contribute to the overall complexity of the particles, could be replaced by these indexes effectively.
doi:10.1186/1475-2859-5-5
PMCID: PMC1382250  PMID: 16472407
19.  Generating Signals with Multiscale Time Irreversibility: The Asymmetric Weierstrass Function 
Complexity  2011;16(4):29-38.
Time irreversibility (asymmetry with respect to time reversal) is an important property of many time series derived from processes in nature. Some time series (e.g., healthy heart rate dynamics) demonstrate even more complex, multiscale irreversibility, such that not only the original but also coarse-grained time series are asymmetric over a wide range of scales. Several indices to quantify multiscale asymmetry have been introduced. However, there has been no simple generator of model time series with “tunable” multiscale asymmetry to test such indices. We introduce an asymmetric Weierstrass function WA (constructed from asymmetric sawtooth functions instead of cosine waves) that can be used to construct time series with any given value of the multiscale asymmetry. We show that multiscale asymmetry appears to be independent of other multiscale complexity indices, such as fractal dimension and multiscale entropy. We further generalize the concept of multiscale asymmetry by introducing time-dependent (local) multiscale asymmetry and provide examples of such time series. The WA function combines two essential features of complex fluctuations, namely fractality (self-similarity) and irreversibility (multiscale time asymmetry); moreover, each of these features can be tuned independently. The proposed family of functions can be used to compare and refine multiscale measures of time series asymmetry.
doi:10.1002/cplx.20333
PMCID: PMC3398483  PMID: 22815622
multiscale time asymmetry; irreversibility; arrow of time; sawtooth function; Weierstrass function; multiscale analysis of time series
20.  Fractal Analysis Reveals Reduced Complexity of Retinal Vessels in CADASIL 
PLoS ONE  2011;6(4):e19150.
The Cerebral Autosomal Dominant Arteriopathy with Subcortical Infarcts and Leukoencephalopathy (CADASIL) affects mainly small cerebral arteries and leads to disability and dementia. The relationship between clinical expression of the disease and progression of the microvessel pathology is, however, uncertain as we lack tools for imaging brain vessels in vivo. Ophthalmoscopy is regarded as a window into the cerebral microcirculation. In this study we carried out an ophthalmoscopic examination in subjects with CADASIL. Specifically, we performed fractal analysis of digital retinal photographs. Data are expressed as mean fractal dimension (mean-D), a parameter that reflects complexity of the retinal vessel branching. Ten subjects with genetically confirmed diagnosis of CADASIL and 10 sex and age-matched control subjects were enrolled. Fractal analysis of retinal digital images was performed by means of a computer-based program, and the data expressed as mean-D. Brain MRI lesion volume in FLAIR and T1-weighted images was assessed using MIPAV software. Paired t-test was used to disclose differences in mean-D between CADASIL and control groups. Spearman rank analysis was performed to evaluate potential associations between mean-D values and both disease duration and disease severity, the latter expressed as brain MRI lesion volumes, in the subjects with CADASIL. The results showed that mean-D value of patients (1.42±0.05; mean±SD) was lower than control (1.50±0.04; p = 0.002). Mean-D did not correlate with disease duration nor with MRI lesion volumes of the subjects with CADASIL. The findings suggest that fractal analysis is a sensitive tool to assess changes of retinal vessel branching, likely reflecting early brain microvessel alterations, in CADASIL patients.
doi:10.1371/journal.pone.0019150
PMCID: PMC3083432  PMID: 21556373
21.  Mos as a tool for genome-wide insertional mutagenesis in Caenorhabditis elegans: results of a pilot study 
Nucleic Acids Research  2004;32(14):e117.
The sequence of the Caenorhabditis elegans genome contains approximately 19 000 genes. Available mutants currently exist for <20% of these genes. The existence of a Mos-based inducible transposon system in C.elegans could theoretically serve as a tool to saturate the genome with insertions. We report here the results of a pilot study aimed at assaying this strategy. We generated 914 independent random Mos insertions and determined their location by inverse PCR. The distribution of the insertions throughout the genome does not reveal any gross distortion, with the exception of a major hotspot on chromosome I (rDNA locus). Transposons are evenly distributed between the genic and intergenic regions. Within genes, transposons insert preferentially into the introns. We derived the consensus target site for Mos in C.elegans (ATATAT), which is common to Tc1, another mariner element. Finally, we assayed the mutagenic properties of insertions located in exons by comparing the phenotype of homozygous strains to that of known mutations or RNAi of the same gene. This pilot experiment shows that a Mos-based approach is a viable strategy that can contribute to the constitution of genome-wide collections of identified C.elegans mutants.
doi:10.1093/nar/gnh111
PMCID: PMC514395  PMID: 15310838
22.  Higuchi Fractal Properties of Onset Epilepsy Electroencephalogram 
Epilepsy is a medical term which indicates a common neurological disorder characterized by seizures, because of abnormal neuronal activity. This leads to unconsciousness or even a convulsion. The possible etiologies should be evaluated and treated. Therefore, it is necessary to concentrate not only on finding out efficient treatment methods, but also on developing algorithm to support diagnosis. Currently, there are a number of algorithms, especially nonlinear algorithms. However, those algorithms have some difficulties one of which is the impact of noise on the results. In this paper, in addition to the use of fractal dimension as a principal tool to diagnose epilepsy, the combination between ICA algorithm and averaging filter at the preprocessing step leads to some positive results. The combination which improved the fractal algorithm become robust with noise on EEG signals. As a result, we can see clearly fractal properties in preictal and ictal period so as to epileptic diagnosis.
doi:10.1155/2012/461426
PMCID: PMC3296200  PMID: 22461844
23.  Key players in chromosome segregation in Caenorhabditis elegans 
In contrast to many eukaryotic organisms in which kinetochores are assembled on localized centromeres of monocentric chromosomes, Caenorhabditis elegans has diffuse kinetochores, termed holo-kinetochores, which are assembled along the entire length of the mitotic chromosome. Despite this cytologically distinct chromosomal architecture, holo-kinetochores of C. elegans and kinetochores of other eukaryotes share structurally and functionally conserved properties. The amphitelic attachment of sister kinetochores to microtubules can be achieved by proper chromosomal organization, which relies on spatiotemporally orchestrated functions of conserved protein complexes such as the cohesin, condensin, and chromosomal passenger complexes during mitosis and meiosis in C. elegans. Moreover, the structure of spindle assembly checkpoint components and their safeguard function are also well conserved in C. elegans. Extensive efforts in the last few years to elucidate the molecular mechanisms of the C. elegans spindle assembly checkpoint have revealed its unique features. In this review, I will focus on the conservation and diversity of proteins that are required to maintain chromosome transmission fidelity during mitosis and meiosis in C. elegans.
PMCID: PMC2745088  PMID: 19273145
C. elegans; Kinetochore; Spindle assembly checkpoint
24.  Conservation of long-range synteny and microsynteny between the genomes of two distantly related nematodes 
Genome Biology  2002;3(10):research0057.1-research0057.14.
To assess whether the pattern of high rates of genome rearrangement, with a bias towards within-chromosome events is true of nematodes in general, genome sequence was used to compare the model Caenorhabditis elegans and the filarial parasite Brugia malayi. It is suggested that intrachromosomal rearrangement is a major force driving chromosomal organization in nematodes.
Background
Comparisons between the genomes of the closely related nematodes Caenorhabditis elegans and Caenorhabditis briggsae reveal high rates of rearrangement, with a bias towards within-chromosome events. To assess whether this pattern is true of nematodes in general, we have used genome sequence to compare two nematode species that last shared a common ancestor approximately 300 million years ago: the model C. elegans and the filarial parasite Brugia malayi.
Results
An 83 kb region flanking the gene for Bm-mif-1 (macrophage migration inhibitory factor, a B. malayi homolog of a human cytokine) was sequenced. When compared to the complete genome of C. elegans, evidence for conservation of long-range synteny and microsynteny was found. Potential C. elegans orthologs for II of the 12 protein-coding genes predicted in the B. malayi sequence were identified. Ten of these orthologs were located on chromosome I, with eight clustered in a 2.3 Mb region. While several, relatively local, intrachromosomal rearrangements have occurred, the order, composition, and configuration of two gene clusters, each containing three genes, was conserved. Comparison of B. malayi BAC-end genome survey sequence to C. elegans also revealed a bias towards intrachromosome rearrangements.
Conclusions
We suggest that intrachromosomal rearrangement is a major force driving chromosomal organization in nematodes, but is constrained by the interdigitation of functional elements of neighboring genes.
PMCID: PMC134624  PMID: 12372145
25.  Classification of surface EMG signal with fractal dimension*  
Surface EMG (electromyography) signal is a complex nonlinear signal with low signal to noise ratio (SNR). This paper is aimed at identifying different patterns of surface EMG signals according to fractal dimension. Two patterns of surface EMG signals are respectively acquired from the right forearm flexor of 30 healthy volunteers during right forearm supination (FS) or forearm pronation (FP). After the high frequency noise is filtered from surface EMG signal by a low-pass filter, fractal dimension is calculated from the filtered surface EMG signal. The results showed that the fractal dimensions of filtered FS surface EMG signals and those of filtered FP surface EMG signals distribute in two different regions, so the fractal dimensions can represent different patterns of surface EMG signals.
doi:10.1631/jzus.2005.B0844
PMCID: PMC1389869  PMID: 16052721
Surface EMG signal; Fractal dimension; Correlation dimension; Self-similarity; GP algorithm

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