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.     

Production and effects of volatile organic compounds during interspecific interactions

Competition between mycelia of saprotrophic cord-forming basidiomycetes occurs both within dead woody resources and in the soil-litter interface, and involves a variety of antagonistic mechanisms including the production of volatile organic compounds (VOCs). The antagonistic potential of VOC profiles from interactions in wood blocks and in soil microcosms was assessed using shared headspace experiments, and the profile of VOCs emitted over the course of interactions elucidated using solid phase microextraction (SPME) with gas chromatography-mass spectrometry (GC–MS). Quantitative and qualitative changes in VOC production occurred in interactions compared to self-pairing controls, with different VOC profiles from fungi growing in wood blocks compared to soil trays. There were both stimulatory and inhibitory effects of VOCs on target mycelial extension rate, hyphal coverage and fractal dimension. VOC-mediated effects were greater in self-pairing controls compared to interactions, and differed depending on the substratum in which the VOC-producing fungi were growing.     

Contrasting spatial heterogeneity of sessile organisms within mussel (Perna perna L.) beds in relation to topographic variability

We examined the spatial heterogeneity in three sessile rocky shore organisms, the mussel Perna perna, the barnacle Octomeris angulosa (Sowerby) and the red alga Gelidium pristoides (Turn.) at a range of continuous local scales along horizontal transects within mid- and upper mussel beds of South African shores. We also examined the relationships between variability of organisms and topographic features (rock depressions, slope, aspect), and between mussel, barnacle and algal variability over the same scales. To estimate spatial heterogeneity, we analyzed scaling properties of semivariograms using a fractal approach. Relationships between different variables at the different scales were examined by cross-semivariograms. Spatial dependence of P. perna variability increased with spatial dependence of topographic variability, so that scaling regions of mussel and topographic distributions corresponded well. This relationship often improved with larger local scales (mussel cover increased with depressions, steeper slope and aspect towards waves), while at smaller spatial scales, variability in mussel cover was less well explained by variability in topography. The variability of the barnacle O. angulosa exhibited spatial dependence, even on topographically unstructured shores. In contrast, the distribution of the alga G. pristoides revealed high fractal dimensions, showing spatial independence on topographically unstructured shores. Algae also showed a very strong negative relationship with mussels at most local scales, and a negative relationship with barnacles in upper zones, especially at larger local scales. Barnacles may show clear spatial dependence because of hydrodynamics (at larger local scales) and the need to find a future mate in close proximity (at smaller local scales), while algae may show a strong negative relationship with mussels because of competition for space.     

后河自然保护区珍稀濒危植物种群分布格局的分形特征：计盒维数

通过4种优势树种和7种珍稀树种种群的计盒维数来反映种群分布格局的分形特征．结果表明,4种优势种群的计盒维数在1．346-1．414之间,在历研究的整个常绿落叶阔叶混交林群落中占据了较大的生态空间;其中尖连蕊茶种群的计盒维数最大,占据最大的生态空间．而7种珍稀树种种群的计量续数除了金钱槭种群大于1之外,其余6个种群的计盒维数都小于1,说明它们占据生态空间的能力较小．并且这11个种群的拐点尺度都出现在5-12．5m之间．根据分形的自相似性,可以推断种群在大于拐点尺度的所有尺度上都为同一分布类型,并且在拐点前后种群的格局类型夏发生变化．种群分布格局的判定结果部分地验证了这一观点,11个物种中有5个物种的种群格局在拐点前后发生变化,另6种没有发生这种变化。     

广西木论自然保护区铁榄群落主要乔木种群格局的分形特征

为了揭示种群空间分布特征随尺度的变化规律,探讨不同种群对空间资源的占据能力和生态适应能力及分形差异,采用计盒维数、信息维数和关联维数对木论自然保护区铁榄群落主要乔木种群格局特征进行了研究。结果表明,优势种铁榄（Sinosideroxylon pedunculatum）、东女贞（Ligustrum japonicum）、小栾树（Boniodendron minius）、稀有濒危种掌叶木（Handeliodendron bodinieri）、青檀（Pteroceltis tatarinowii）5个种群在相应的尺度具有明显的分形特征;计盒维数和信息维数大小顺序一致,均为：铁榄〉东女贞〉小栾树〉掌叶木〉青檀;在5个种群中,铁榄计盒维数为1．7763最大,接近2,对空间环境的占据能力最强,信息维数（1．7206）也最大,个体聚集强度强,在群落中处于绝对优势地位,青檀计盒维数（0．8316）和信息维数（0．8990）最小,对环境的占据利用能力弱,个体聚集程度整体较弱,在群落中处于伴生地位;在0．5～50m的尺度上,关联维数大小顺序为：铁榄（1．7314）〉东女贞（1．6688）〉小栾树（1．6050）〉青檀（0．7868）,而掌叶木只有在大的尺度上才出现关联,且关联较强。3个分形维数大小的不同组合,能够全面揭示不同种群格局特征和种群在群落中所处地位的差异。     

On the importance of body size in the colonisation of ephemeral resource patches by vagile consumers

A study on the colonisation of leaf detritus patches by vagile macroinvertebrates in a brackish lagoon is presented in the framework of a conceptual model where a body size-related constraint on patch use behaviour is explicitly considered. Abundance patterns of dominant macroinvertebrate taxa were characterised by short-term, non-random fluctuations, showing significant site-dependent variations. Yet, a site-independent covariation was observed between patterns’ fractal dimension and the average body mass of each taxon, indicating that, while the temporal scales characterising the colonisation patterns may be highly species-specific, cross-species generalisations are possible based on body size. The generality of these results was supported by literature data on temporal patterns of carcass colonisation by bathyal fish. The importance of size-related mechanisms in regulating the aggregation of vagile consumers on resource patches and, ultimately, their coexistence at both an inter- and intra-specific level, is discussed.     

Morphostructural constraints and phylogenetic overprint on sutural frilling in Late Jurassic ammonites

The functional significance of frilled septa and complex sutures in ammonoids has generated ongoing debate. The 'classic' hypothesis envisages ammonoid shells and septa as designed for resisting ambient hydrostatic pressure, complex sutures being the evidence of strength in shells for colonization of deep habitats. Here we address the 'suture problem', focusing on the analysis and interpretation of variables in our database of Late Jurassic ammonites not included in previous studies, such as whorl height ( W h ), whorl shape ( S ), shell coiling ( WD ), taxonomic grouping and basic planispiral shell shape. The results indicate that sutural complexity, as measured by the fractal dimension ( D f ) value of the suture line, is positively correlated with W h , and that the sutures of oceanic shells tend to provide, for a given W h value, lower D f estimates than do those of neritic shells. No general trend of increase in sutural complexity was noted for specimens recovered from swell areas belonging to oceanic fringes with respect to those that inhabited epicontinental shelves. In fact, Perisphinctoidea, the clade best represented in the database analysed, shows a higher D f mean value in neritic species than in epioceanic ones. Significant differences in sutural complexity were detected for groups of ammonites classified according to shell shapes ( WD , S ). Oxycones and discocones, streamlined potential swimmers, show the highest D f mean values, while spherocones and cadicones, which were presumably vertical vagrants, present the lowest ones. This indicates that sutural complexity was more related to shell geometry than to bathymetry.     

Temporal pattern of locomotor activity in Drosophila melanogaster

The temporal pattern of locomotor activity of single Drosophila melanogaster flies freely walking in small tubes is described. Locomotor activity monitored by a light gate has a characteristic time-course that depends upon age and the environmental conditions. Several methods are applied to assess the complexity of the temporal pattern. The pattern varies according to sex, genotype, age and environmental conditions (food; light). Activity occurs clustered in bouts. The intrinsic bout structure is quantified by four parameters: number of light gate passages (counts) per bout, duration of a bout, pause between two successive bouts and mean bout period. In addition, the distribution of the periods between light-gate crossings (inter-count intervals) as function of inter-count interval duration reveals a power law, suggesting that the overall distribution of episodes of activity and inactivity has a fractal structure. In the dark without food, the fractal dimension which represents a measure of the complexity of the pattern is sex, genotype and age specific. Fractality is abolished by additional sensory stimulation (food; light). We propose that time-course, bout structure and fractal dimension of the temporal pattern of locomotor activity describe different aspects of the fly's central pattern generator for locomotion and its motivational control. Accepted: 10 October 1998     

Fractal dimension analysis and mathematical morphology of structural changes in actin filaments imaged by electron microscopy

In this work, we examined structural changes of actin filaments interacting with myosin visualized by quick freeze deep-etch replica electron microscopy (EM) by using a new method of image processing/analysis based on mathematical morphology.In order to quantify the degree of structural changes, two characteristic patterns were extracted from the EM images. One is the winding pattern of the filament shape (WP) reflecting flexibility of the filament, and the other is the surface pattern of the filament (SP) reflecting intra-molecular domain-mobility of actin monomers constituting the filament. EM images were processed by morphological filtering followed by box-counting to calculate the fractal dimensions for WP (D_WP) and SP (D_SP). The result indicates that D_WP was larger than D_SP irrespective of the state of the filament (myosin-free or bound) and that both parameters for myosin-bound filaments were significantly larger than those for myosin-free filaments. Overall, this work provides the first quantitative insight into how conformational disorder of actin monomers is correlated with the myosin-induced increase in flexibility of actin filaments along their length as suggested by earlier studies with different techniques. Our method is yet to be improved in details, but promising as a powerful tool for studying the structural change of protein molecules and their assemblies, which can potentially be applied to a wide range of biological and biomedical images.     

Genotypic variation in seedling root architectural traits and implications for drought adaptation in wheat (Triticum aestivum L.)

Root system characteristics are of fundamental importance to soil exploration and below-ground resource acquisition. Root architectural traits determine the in situ space-filling properties of a root system or root architecture. The growth angle of root axes is a principal component of root system architecture that has been strongly associated with acquisition efficiency in many crop species. The aims of this study were to examine the extent of genotypic variability for the growth angle and number of seminal roots in 27 current Australian and 3 CIMMYT wheat (Triticum aestivum L.) genotypes, and to quantify using fractal analysis the root system architecture of a subset of wheat genotypes contrasting in drought tolerance and seminal root characteristics. The growth angle and number of seminal roots showed significant genotypic variation among the wheat genotypes with values ranging from 36 to 56 (degrees) and 3 to 5 (plant⁻¹), respectively. Cluster analysis of wheat genotypes based on similarity in their seminal root characteristics resulted in four groups. The group composition reflected to some extent the genetic background and environmental adaptation of genotypes. Wheat cultivars grown widely in the Mediterranean environments of southern and western Australia generally had wider growth angle and lower number of seminal axes. In contrast, cultivars with superior performance on deep clay soils in the northern cropping region, such as SeriM82, Baxter, Babax, and Dharwar Dry exhibited a narrower angle of seminal axes. The wheat genotypes also showed significant variation in fractal dimension (D). The D values calculated for the individual segments of each root system suggested that, compared to the standard cultivar Hartog, the drought-tolerant genotypes adapted to the northern region tended to distribute relatively more roots in the soil volume directly underneath the plant. These findings suggest that wheat root system architecture is closely linked to the angle of seminal root axes at the seedling stage. The implications of genotypic variation in the seminal root characteristics and fractal dimension for specific adaptation to drought environment types are discussed with emphasis on the possible exploitation of root architectural traits in breeding for improved wheat cultivars for water-limited environments.