Structural complexity in mussel beds: the fractal geometry of surface topography

Seafloor topographic complexity is ecologically important because it provides habitat structure and alters boundary-layer flow over the bottom. Despite its importance, there is little agreement on how to define and measure surface complexity. The purpose of this investigation was to utilize fractal geometry of vertical cross-section profiles to characterize the surface topography of the soft-bottom mussel bed (Mytilus edulis L.) at Bob's Cove, ME, USA. Mussels there have been shown previously to have spatially ordered fractal characteristics in the horizontal plane. Two hypotheses were tested. The first was that the bed surface is fractal over the spatial scale of 1.44-200 mm, with fractal dimension less than or equal to 1.26, the value for the Koch curve, our model for bed profiles. The second was that bed surface topography (i.e., in vertical profile) is less complex than the mussel bed spatial pattern (i.e., aerial view in the horizontal plane). Both hypotheses were supported. Cross-sections of plaster casts of the bed produced 88 surface profiles, all of which were fractal over the entire spatial scale of more two orders of magnitude employed in the analysis. Fractal dimension values (D) for individual profiles ranged from 1.031 to 1.310. Fractal dimensions of entire casts ranged up to mean (1.242+/-0.046) and median (1.251) values similar to 1.26, the theoretical value of the Koch curve. The bed surface was less complex than the bed spatial pattern because every profile had D<1.36, the smallest value previously obtained from aerial views of the bed. The investigation demonstrated for the first time that surface topography of a soft-bottom mussel bed was fractal at a spatial scale relevant to hydrodynamic processes and habitat structure important for benthic organisms. The technique of using cross-section profiles from casts of the bed surface avoided possible underestimates of fractal dimension that can result from other profiling methods reported in the literature. The results demonstrate that fractal dimension can be useful in the analysis of habitat space and water flow over any irregular seafloor surface because it incorporates the size, shape, and scale of roughness elements into a simple, numerical metric.     

Fat fractal and multifractals for protein and enzyme surfaces.

The roughness and irregularity of the surfaces in the protein and enzyme are fractal features that may be characterized by fractal dimensions and mass exponents. The surface fractal dimensions calculated by the variation method are different from those obtained by other methods, since the former is applicable to the self-affine system. Thus the results reported here are reliable for the surfaces. However, the fat fractal and multifractal features of proteins and enzymes are studied by simulation. The surface mass exponents are regarded as another kind of scaling exponent, and the spectrum f(alpha) provides further detailed information about the surfaces of enzyme and protein. The applications of the spectrum f(alpha) to the enzymatic reactions is also discussed.     

原子力显微镜对壳聚糖分形结构的研究

利用原子力显微镜（Atomic Force Microscope,AFM)研究了壳聚糖分子在云母表面的分形聚集生长过程,并对壳聚糖分形生长的作用机理及其生长过程进行研究,发现壳聚糖分子在聚集生长过程呈传统的具有分形特征的正态分布和奇异分布。在壳聚糖聚集生长过程中,由于溶液的自然挥发,形成了云母基片中心位置的壳聚糖浓度相对较高,而周围壳聚糖的浓度相对较低的阶梯分布,因此呈现出中部的胶粒大而周围的胶粒较小的现象。AFM图像显示在云母片的中心部位壳聚糖分子聚集生长为“树”形结构而在边缘部位呈“星”形结构,这两种结构都具有典型的自相似性,壳聚糖的分形生长与其计算机模拟树形模式和DLCA模式拟合得很好。     

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.     

Trends in soil fractal parameters caused by accumulation of soil organic matter as resulting from the analysis of water vapor adsorption isotherms

The experimental adsorption isotherms are used to evaluate the specific surface area and the surface fractal dimensions of several samples containing organic matter. The aim of the investigations has been to search for correlations between specific surface area and geometrical heterogeneity, as characterized by the surface fractal dimension and the content of organic matter. Besides natural organic soils (peats, forest humuses and brown coal from mines) we also investigate controlled systems obtained by adding humic acids to kaolin and quartz, organic compost to soils and by mixing sandy soil and peat at different ratios. The aim of the investigations of controlled systems was to discover some general trends in the dependencies of the specific surface area and the surface fractal dimension on the content of organic matter.     

Fractal dimension analysis of factor X activation in the presence of tissue factor-factor viia complex in a continuous flow reactor

Summary The characteristics of a phospholipid surface are of major importance in the activation of factor X in the presence of tissue factor-factor VIIa (TF-VIIa) complex. A possible tool which provides a measure of the surface corrugation and roughness is the fractal dimension analysis. This paper uses the fractal characterization of a phospholipid surface to develop a model for analyzing surface based enzymatic reaction data. The modeling indicates that the fractal dimension (D) of a phospholipid surface is a function of the wall shear rate. The results also indicate that the fractal dimension of the phospholipid surface decreases from approximately 2.9 to 1.4 as the wall shear rate increases from 50 to 1600 sec^–1. At the same time the factor Xa production increases from 1.9 to 5.8 pmoles/ (min. cm^–2).The results of the fractal dimension analysis clearly indicates that the surface roughness of a phospholipid surface may have a significant effect on factor X activation.     

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.     

Nonlinear kinetics of ferritin adsorption.

The adsorption of ferritin at a methylized quartz surface was measured with off-null ellipsometry and transmission electron microscopy. An initial lag-phase was seen, followed by an accelerating adsorption leading to mass transport limitation of the reaction. The rate of adsorption then decreased at a surface concentration far below monolayer coverage, and a continuously decreasing rate of binding was seen. The slope of the binding rate was linear with the logarithm of time (fractal kinetics). The adsorbed ferritin molecules were distributed in clusters as seen by transmission electron microscopy. Clusters grown during the mass transport limited adsorption had crystalline structure at short range and low fractal dimensions (df = 0.89) over long range. Clusters grown during adsorption with fractal kinetics showed random structure at short range and a high fractal dimension df = 1.86 over all ranges. These findings indicate some new important mechanisms responsible for the complex kinetics of macromolecular reactions at solid-liquid interfaces. The results are discussed in relation to recently developed theories of self-organized criticality.     

基于分形理论的木材顺纹理断裂研究

研究5种木材顺纹理断裂性质和断面分形特征,测量断口表面的分形维数,建立了分形维数与断裂韧性间的关系.研究结果表明：（1）不同树种因其构造差异对裂纹扩展的阻力不同,这种差异同样也表现在材料断裂表面的形貌特征上;（2）木材顺纹理断裂韧性K（IC）^（TL）与分形维数D之间有很高的正比关系：y=0.036x＋2.162,（R^2=0.98）,揭示了断口分形维数与材料性能之间的内在联系.     

Surface fractality as a guide for studying protein-protein interactions

The concept of fractal dimension is applied to protein surfaces. Satellite tobacco necrosis virus, prealhumin, retinol binding protein and lysozyme have been studied. A residue fractal index has been defined, which provides a suitable colour code when using computer graphics for visualizing surfaces. Some provisions are made that render the MS algorithm useful to calculate protein surface fractal dimensions. It has been found that a correlation exists between regions of high fractal dimension and those involved in protein-protein interactions. The usefulness of surface fractality in this context is demonstrated by a molecular docking experiment.     

The fractal geometry of nutrient exchange surfaces does not provide an explanation for 3/4-power metabolic scaling

Background  

A prominent theoretical explanation for 3/4-power allometric scaling of metabolism proposes that the nutrient exchange surface of capillaries has properties of a space-filling fractal. The theory assumes that nutrient exchange surface area has a fractal dimension equal to or greater than 2 and less than or equal to 3 and that the volume filled by the exchange surface area has a fractal dimension equal to or greater than 3 and less than or equal to 4.     

A single- and a dual-fractal analysis of antigen-antibody binding kinetics for different biosensor applications.

The diffusion-limited binding kinetics of antigen (or antibody) in solution to antibody (or antigen) immobilized on a biosensor surface is analyzed within a fractal framework. The data is adequately described by a single- or a dual-fractal analysis. Initially, the data was modelled by a single-fractal analysis. If an inadequate fit was obtained then a dual-fractal analysis was utilized. The regression analysis provided by Sigmaplot, 1993 (Scientific Graphing Software: User's Manual. Jandel Scientific, San Rafael, CA) was utilized to determine if a single-fractal analysis is sufficient, or a dual-fractal analysis is required. In general, it is of interest to note that the binding rate coefficient and the fractal dimension exhibit changes in the same direction (except for a single example) for the antigen-antibody systems analyzed. Binding rate coefficient expressions as a function of the fractal dimension developed for the antigen-antibody binding systems indicate a high sensitivity of the binding rate coefficient on the fractal dimension when both a single -as well as a dual-fractal analysis is used. For example, for a single-fractal analysis and for the binding of human endothelin-1 (ET-1) antibody in solution to ET-1(15-21) x BSA (bovine serum albumin) immobilised on a surface plasmon resonance surface, the order of dependence of the binding rate coefficient, k on the fractal dimension, Df is 7.0945. Similarly, for a dual-fractal analysis and for the binding of parasite L. donovani diluted pooled sera in solution to fluorescein isothiocyanate-labeled anti-human immunoglobulin IgG immobilized on an optical fibre, the order of dependence of k1 and k2 on Df1 and Df2 were 6.8018 and -4.393, respectively. Binding rate coefficient expressions are also developed as a function of the analyte (antigen or antibody) concentration in solution. The binding rate coefficient expressions developed as a function of the fractal dimension(s) are of particular value since they provide a means to better control biosensor performance by linking it to the heterogeneity on the surface, and emphasize in a quantitative sense the importance of the nature of the surface in biosensor performance.     

Fractal dimensions and porosities of Zoogloea ramigera and Saccharomyces cerevisae aggregates

The fractal nature microbial aggregates is a function of the type of microorganism and mixing conditions used to develop aggregates. We determined fractal dimensions from length-projected area (D(2)) and length-number scaling (D(3)) relationships. Aggregates of Zoogloea ramigera developed in rotating test tubes were both surface and mass fractals, with fractal dimensions of D(2) = 1.69 +/- 0.11 and D(3)= 1.79 +/- 0.28 (+/-standard deviation), respectively. When we grew this bacteria in a bench-top fermentor, aggregates maintained their surface fractal characteristics (D(2) = 1.78 +/- 0.11) but lost their mass fractal characteristics (D(3) = 2.99 +/- 0.36). Yeast aggregates (Saccharomyces cerevisae) grown in rotating tests tubes had higher average fractal dimensions than bacterial aggregates grown under physically identical conditions, and were also considered fractal (D(2) = 1.92 +/- 0.08 and D(3) = 2.66 +/- 0.34). Aggregates porosity can be expressed in term of a fractal dimensions, but average porosities are higher than expected. The porosities of yeast aggregates (0.9250-0.9966) were similar to porosities of bacterial aggregates (0.9250-0.9966) cultured under the same physical conditions, although bacterial aggregates developed in the reactor had higher average porosities (0.9857-0.9980). These results suggest that that scaling relationships based on fractal geometry may be more useful than equations derived from Euclidean geometry for quantifying the effects of different fluid mechanical environments on aggregates morphology and characteristics such as density, porosity, and projected surface area.     

川西亚高山针叶林土壤颗粒的分形特征

分形理论为土壤等复杂体系的定量化研究提供了一种有效工具。本文以12个川西亚高山针叶林土壤颗粒组成数据为基础,运用分形模型研究了亚高山针叶林土壤颗粒的分形维数。结果表明,12个川西亚高山针叶林表层土壤颗粒的分形维数D为2.5209~2.7978。通过逐步多元回归分析,土壤颗粒分形维数仅与<0.001 mm颗粒含量的相关系数达极显著。土壤颗粒分形维数与土壤有机质含量、全氮含量和pH相关性不显著,然而土壤有机质含量与全氮含量呈极显著正相关。本研究探讨了利用土壤颗粒粒径分布的分形维数来定量表征川西亚高山针叶林土壤的特征。     

Three-Dimensional Surface Parameters and Multi-Fractal Spectrum of Corroded Steel

To study multi-fractal behavior of corroded steel surface, a range of fractal surfaces of corroded surfaces of Q235 steel were constructed by using the Weierstrass-Mandelbrot method under a high total accuracy. The multi-fractal spectrum of fractal surface of corroded steel was calculated to study the multi-fractal characteristics of the W-M corroded surface. Based on the shape feature of the multi-fractal spectrum of corroded steel surface, the least squares method was applied to the quadratic fitting of the multi-fractal spectrum of corroded surface. The fitting function was quantitatively analyzed to simplify the calculation of multi-fractal characteristics of corroded surface. The results showed that the multi-fractal spectrum of corroded surface was fitted well with the method using quadratic curve fitting, and the evolution rules and trends were forecasted accurately. The findings can be applied to research on the mechanisms of corroded surface formation of steel and provide a new approach for the establishment of corrosion damage constitutive models of steel.     

黑土表层土壤颗粒的分形特征

基于第二次全国土壤普查结果,应用土壤颗粒的质量分布计算了36个典型剖面表层土壤颗粒的分形维数值.结果表明:土壤颗粒分形维数值D在2.5831～2.8230,其变异性极弱,且分形维数值随质地变细而增大;土壤机械组成中,砂粒(2～0.02mm)含量、粉粒(0.02～0.002mm)含量与分形维数值均呈显著负相关(P<0.05);粘粒(<0.002mm)含量与土壤分形维数值呈极显著正相关(P<0.01);分形维数值D与土壤中的有机质、全N、全P、全K含量及pH值相关性均不显著.土壤分布的分形维数可以作为反映黑土退化程度的一个综合性定量指标.     

Microstructural characterization of chitosan and alginate films by microscopy techniques and texture image analysis

The aim of this work is to characterize the microstructure of chitosan and alginate edible films by microscopy techniques and texture image analysis. Edible films were obtained by solution casting and solvent evaporation. The microscopy techniques used in this work were: light, environmental scanning electron and atomic force microscopy. Textural features and fractal dimension were extracted from the images. Entropy and fractal dimension were more useful to evaluate the complexity and roughness of films. The highest values of entropy and fractal dimension corresponded to alginate/chitosan, followed of alginate and chitosan films. An entropy/fractal dimension ratio, proposed here, was useful to characterize the degree of image complexity and roughness of edible films at different magnifications. It was possible to postulate that microscopy techniques combined with texture image analysis are efficient tools to quantitatively evaluate the surface morphology of edible films made of chitosan and alginate.     

长江上游暗针叶林优势树种峨嵋冷杉的树体分维结构研究

以长江上游亚高山地区水碛物、坡积物上的原始峨嵋冷杉为研究对象,采用分形几何理论剖析了树体分枝结构,结果表明,峨嵋冷杉树冠的最大冠幅出现在树冠下部的20％处,其体积可以用圆台与圆锥来模拟,主干与侧枝分枝均长短相间排列,树冠表面积和侧枝发枝结构具有分形分布特征,分形维数分别为2．3232和1．8119,相关系达0．99。     

A model for enhanced nucleation of protein crystals on a fractal porous substrate

The phenomenon of enhanced nucleation and crystallization of proteins on porous silicon (PS) is theoretically studied and explained. The PS layer is treated as a fractal structure, and a new mechanism of local supersaturation associated with the fractality is proposed. It is shown that the number of adsorbed molecules on a fragment with a fractal surface significantly exceeds that on one with flat surfaces. For a fractal PS surface, a local concentration of molecules that is sufficient for nucleation is possible inside and in the close vicinity of the pores, even when the average conditions in the bulk of the solution correspond to metastability. The wide distribution of fractal pore size is favorable for the crystallization of a wide range of macromolecules using the same sample. In addition, the PS technology is very flexible, allowing tailoring the pore size and concentration as well as the fractal properties to specific proteins by changing the fabrication conditions.