Power-law versus exponential distributions of animal group sizes

There has been some confusion concerning the animal group size: an exponential distribution was deduced by maximizing the entropy; lognormal distributions were practically used; as power-law decay with exponent 3/2 was proposed in physical analogy to aerosol condensation. Here I show that the animal group-size distribution follows a power-law decay with exponent 1, and is truncated at a cut-off size which is the expected size of the groups an arbitrary individual engages in. An elementary model of animal aggregation based on binary splitting and coalescing on contingent encounter is presented. The model predicted size distribution holds for various data from pelagic fishes and mammalian herbivores in the wild.     

Modeling fractal structure of city-size distributions using correlation functions

Zipf's law is one the most conspicuous empirical facts for cities, however, there is no convincing explanation for the scaling relation between rank and size and its scaling exponent. Using the idea from general fractals and scaling, I propose a dual competition hypothesis of city development to explain the value intervals and the special value, 1, of the power exponent. Zipf's law and Pareto's law can be mathematically transformed into one another, but represent different processes of urban evolution, respectively. Based on the Pareto distribution, a frequency correlation function can be constructed. By scaling analysis and multifractals spectrum, the parameter interval of Pareto exponent is derived as (0.5, 1]; Based on the Zipf distribution, a size correlation function can be built, and it is opposite to the first one. By the second correlation function and multifractals notion, the Pareto exponent interval is derived as [1, 2). Thus the process of urban evolution falls into two effects: one is the Pareto effect indicating city number increase (external complexity), and the other the Zipf effect indicating city size growth (internal complexity). Because of struggle of the two effects, the scaling exponent varies from 0.5 to 2; but if the two effects reach equilibrium with each other, the scaling exponent approaches 1. A series of mathematical experiments on hierarchical correlation are employed to verify the models and a conclusion can be drawn that if cities in a given region follow Zipf's law, the frequency and size correlations will follow the scaling law. This theory can be generalized to interpret the inverse power-law distributions in various fields of physical and social sciences.     

Space-irrelevant scaling law for fish school sizes

Universal scaling in the power-law size distribution of pelagic fish schools is established. The power-law exponent of size distributions is extracted through the data collapse. The distribution depends on the school size only through the ratio of the size to the expected size of the schools an arbitrary individual engages in. This expected size is linear in the ratio of the spatial population density of fish to the breakup rate of school. By means of extensive numerical simulations, it is verified that the law is completely independent of the dimension of the space in which the fish move. Besides the scaling analysis on school size distributions, the integrity of schools over extended periods of time is discussed.     

Correlation of SEC/MALLS with ultracentrifuge and viscometric data for chitosans

Attempts have been made to correlate estimates of molecular weight for a group of cationic polysaccharides known as chitosans between the highly popular technique of size-exclusion chromatography coupled to multi-angle laser light scattering, "SEC-MALLS", and the less convenient but more established technique of sedimentation equilibrium in the analytical ultracentrifuge. Four pharmaceutical grade chitosans of various molecular weights and degrees of acetylation (4-30%) were chosen. Better correlation than previous was achieved, although some batch variability was observed. Despite the broad spectrum in degree of acetylation, a log s degrees(20,w) versus log Mw scaling plot appeared to fit a straight line with power-law exponent b=0.25 +/- 0.04, i.e. between the limits of rod (0.15) and coil (0.4-0.5), although this may be the average of a lower b value at low Mw and higher b at high Mw. With regard to viscosity, a logeta versus logMw scaling plot appeared to also fit a straight line with power-law exponent a=0.96 +/- 0.10, again between the coil (0.5-0.7) and rod (1.8) limits.     

7种木本植物的分支指数与代谢指数

West、Brown和Enquist提出的植物分形网络模型(简称WBE模型)认为: 植物的分支指数(1/a, 1/b)决定植物的代谢指数, 当分支指数1/a、1/b分别为理论值2.0、3.0时, 代谢速率与个体大小的3/4次幂成正比, 但是恒定的3/4代谢指数并不能全面地反映植物的代谢情况。基于分支指数的协同变化, Price、Enquist和Savage对WBE模型进行扩展, 提出植物分支参数协同变化模型(简称PES模型)。该文借助于PES模型分析了7种木本植物的分支指数和代谢指数。结果表明: 物种间叶面积与叶生物量呈异速生长关系, 基于叶面积得到的分支指数1/a和代谢指数θ在物种间无显著差异, 基于叶生物量得到的分支指数1/a、1/b和代谢指数θ在物种间均存在显著差异, 但基于叶面积和叶生物量分别拟合出的整体分支指数1/a、1/b和代谢指数θ与理论值均无显著差异, 且用叶面积作为代谢速率的替代指标比用叶生物量分析得出的代谢指数与理论值更接近。今后研究应当关注植物叶面积与叶生物量的异速生长关系对植物代谢速率及相关功能特性的影响。     

Two Fundamental Mechanisms Govern the Stiffening of Cross-linked Networks

Biopolymer networks, such as those constituting the cytoskeleton of a cell or biological tissue, exhibit a nonlinear strain-stiffening behavior when subjected to large deformations. Interestingly, rheological experiments on various in vitro biopolymer networks have shown similar strain-stiffening trends regardless of the differences in their microstructure or constituents, suggesting a universal stiffening mechanism. In this article, we use computer simulations of a random network comprised of cross-linked biopolymer-like fibers to substantiate the notion that this universality lies in the existence of two fundamental stiffening mechanisms. After showing that the large strain response is accompanied by the development of a stress path, i.e., a percolating path of axially stressed fibers and cross-links, we demonstrate that the strain stiffening can be caused by two distinctly different mechanisms: 1) the pulling out of stress-path undulations; and 2) reorientation of the stress path. The former mechanism is bending-dominated and can be recognized by a power-law dependence with exponent 3/2 of the shear modulus on stress, whereas the latter mechanism is stretching-dominated and characterized by a power-law exponent 1/2. We demonstrate how material properties of the constituents, as well as the network microstructure, can affect the transition between the two stiffening mechanisms and, as such, control the dominant power-law scaling behavior.     

Branching and metabolic exponents in seven woody plants

West、Brown和Enquist提出的植物分形网络模型(简称WBE模型)认为: 植物的分支指数(1/a, 1/b)决定植物的代谢指数, 当分支指数1/a、1/b分别为理论值2.0、3.0时, 代谢速率与个体大小的3/4次幂成正比, 但是恒定的3/4代谢指数并不能全面地反映植物的代谢情况。基于分支指数的协同变化, Price、Enquist和Savage对WBE模型进行扩展, 提出植物分支参数协同变化模型(简称PES模型)。该文借助于PES模型分析了7种木本植物的分支指数和代谢指数。结果表明: 物种间叶面积与叶生物量呈异速生长关系, 基于叶面积得到的分支指数1/a和代谢指数θ在物种间无显著差异, 基于叶生物量得到的分支指数1/a、1/b和代谢指数θ在物种间均存在显著差异, 但基于叶面积和叶生物量分别拟合出的整体分支指数1/a、1/b和代谢指数θ与理论值均无显著差异, 且用叶面积作为代谢速率的替代指标比用叶生物量分析得出的代谢指数与理论值更接近。今后研究应当关注植物叶面积与叶生物量的异速生长关系对植物代谢速率及相关功能特性的影响。     

The Evolution of the Exponent of Zipf's Law in Language Ontogeny

It is well-known that word frequencies arrange themselves according to Zipf''s law. However, little is known about the dependency of the parameters of the law and the complexity of a communication system. Many models of the evolution of language assume that the exponent of the law remains constant as the complexity of a communication systems increases. Using longitudinal studies of child language, we analysed the word rank distribution for the speech of children and adults participating in conversations. The adults typically included family members (e.g., parents) or the investigators conducting the research. Our analysis of the evolution of Zipf''s law yields two main unexpected results. First, in children the exponent of the law tends to decrease over time while this tendency is weaker in adults, thus suggesting this is not a mere mirror effect of adult speech. Second, although the exponent of the law is more stable in adults, their exponents fall below 1 which is the typical value of the exponent assumed in both children and adults. Our analysis also shows a tendency of the mean length of utterances (MLU), a simple estimate of syntactic complexity, to increase as the exponent decreases. The parallel evolution of the exponent and a simple indicator of syntactic complexity (MLU) supports the hypothesis that the exponent of Zipf''s law and linguistic complexity are inter-related. The assumption that Zipf''s law for word ranks is a power-law with a constant exponent of one in both adults and children needs to be revised.     

Cluster size distributions: signatures of self-organization in spatial ecologies

Three different lattice-based models for antagonistic ecological interactions, both nonlinear and stochastic, exhibit similar power-law scalings in the geometry of clusters. Specifically, cluster size distributions and perimeter-area curves follow power-law scalings. In the coexistence regime, these patterns are robust: their exponents, and therefore the associated Korcak exponent characterizing patchiness, depend only weakly on the parameters of the systems. These distributions, in particular the values of their exponents, are close to those reported in the literature for systems associated with self-organized criticality (SOC) such as forest-fire models; however, the typical assumptions of SOC need not apply. Our results demonstrate that power-law scalings in cluster size distributions are not restricted to systems for antagonistic interactions in which a clear separation of time-scales holds. The patterns are characteristic of processes of growth and inhibition in space, such as those in predator-prey and disturbance-recovery dynamics. Inversions of these patterns, that is, scalings with a positive slope as described for plankton distributions, would therefore require spatial forcing by environmental variability.     

Influence of microbial concentration on the rheology of non-Newtonian fermentation broths

The objective of this study was to quantify the effect of fungal biomass concentration on the rheology of non-Newtonian fermentation systems. Batch fermentations of Penicillium chrysogenum were carried out with glucose as the sole carbon source. The flow behavior of the system was characterized at various fermentation times and was adequately described by the power-law model. The apparent viscosity of the fermentation broth was significantly affected by biomass concentrations in the fermenter. Fermentation broths containing 17.71 g/l biomass as dry weight were characterized by an apparent viscosity of 0.25 Pa s at a shear rate of 50 s⁻¹. Microbial concentration also affected the power-law flow-behavior index and the consistency index. The value of the consistency index ranged from 0.002 Pa s ⁿ at a biomass concentration of 0.1 g/l to 6.14 Pa s ⁿ at a biomass concentration of 17.71 g/l. The flow-behavior index decreased from an initial value of 1 to a final value of 0.17. Simple empirical correlations have been proposed to quantify the dependence of the power-law parameters on fungal biomass concentration. Experimental data obtained in this study were accurately described by these correlations. The general applicability of these relationships was tested, using previously published rheological data on Aspergillus awamori and Aspergillus niger fermentation broths, and good agreement was seen between experimental data and the predictions from the empirical correlations. Received: 24 March 1998 / Received revision: 10 September 1998 / Accepted: 16 October 1998     

Duplication models for biological networks.

Are biological networks different from other large complex networks? Both large biological and nonbiological networks exhibit power-law graphs (number of nodes with degree k, N(k) approximately k(-beta)), yet the exponents, beta, fall into different ranges. This may be because duplication of the information in the genome is a dominant evolutionary force in shaping biological networks (like gene regulatory networks and protein-protein interaction networks) and is fundamentally different from the mechanisms thought to dominate the growth of most nonbiological networks (such as the Internet). The preferential choice models used for nonbiological networks like web graphs can only produce power-law graphs with exponents greater than 2. We use combinatorial probabilistic methods to examine the evolution of graphs by node duplication processes and derive exact analytical relationships between the exponent of the power law and the parameters of the model. Both full duplication of nodes (with all their connections) as well as partial duplication (with only some connections) are analyzed. We demonstrate that partial duplication can produce power-law graphs with exponents less than 2, consistent with current data on biological networks. The power-law exponent for large graphs depends only on the growth process, not on the starting graph.     

A theoretical interpretation of length-biomass allometry of predominantly bidimensional seaweeds

Experimental studies on bidimensional seaweeds revealed a scaling exponent of 0.472 for their length-biomass allometry. This was significantly higher than the value 0.25, which was proposed earlier as universal for all primary producers, based on the data for unicellular microalgae and vascular plants. Later, an exponent of 0.5 was theoretically derived, which agreed, to some extent, with experimental findings. Here, it is shown that there exists a power-law relation between the two perpendicular length parameters along the directions of growth of a bidimensional organism. The length-biomass allometric parameters can be expressed in terms of this power index. A relation between the allometric scaling exponent and allometric constant, involving the mass per unit area, has been obtained analytically. A method is proposed to determine the power index experimentally. Some mathematical expressions, relating mass, length and other parameters, have been formulated and these would be useful for experimental purposes in allometric studies. Analyzing images from an experimental study, a lot of parameters, regarding flat seaweeds, have been determined by analytical and numerical techniques.     

Network dynamics of the nonlinear power-law relaxation of cell cortex

Living cells are known to exhibit universal power-law rheological behaviors, but their underlying biomechanical principles are still not fully understood. Here, we present a network dynamics picture to decipher the nonlinear power-law relaxation of cortical cytoskeleton. Under step strains, we present a scaling relation between instantaneous differential stiffness and external stress as a result of chain reorientation. Then, during the relaxation, we show how the scaling law theoretically originates from an exponential form of cortical disorder, with the scaling exponent decreased by the imposed strain or crosslinker density in the nonlinear regime. We attribute this exponent variation to the molecular realignment along the stretch direction or the transition of network structure from in-series to in-parallel modes, both solidifying the network toward our one-dimensional theoretical limit. In addition, the rebinding of crosslinkers is found to be crucial for moderating the relaxation speed under small strains. Together with the disorder nature, we demonstrate that the structural effects of networks provide a unified interpretation for the nonlinear power-law relaxation of cell cortex, and may help to understand cell mechanics from the molecular scale.     

Neuronal Spike Timing Adaptation Described with a Fractional Leaky Integrate-and-Fire Model

The voltage trace of neuronal activities can follow multiple timescale dynamics that arise from correlated membrane conductances. Such processes can result in power-law behavior in which the membrane voltage cannot be characterized with a single time constant. The emergent effect of these membrane correlations is a non-Markovian process that can be modeled with a fractional derivative. A fractional derivative is a non-local process in which the value of the variable is determined by integrating a temporal weighted voltage trace, also called the memory trace. Here we developed and analyzed a fractional leaky integrate-and-fire model in which the exponent of the fractional derivative can vary from 0 to 1, with 1 representing the normal derivative. As the exponent of the fractional derivative decreases, the weights of the voltage trace increase. Thus, the value of the voltage is increasingly correlated with the trajectory of the voltage in the past. By varying only the fractional exponent, our model can reproduce upward and downward spike adaptations found experimentally in neocortical pyramidal cells and tectal neurons in vitro. The model also produces spikes with longer first-spike latency and high inter-spike variability with power-law distribution. We further analyze spike adaptation and the responses to noisy and oscillatory input. The fractional model generates reliable spike patterns in response to noisy input. Overall, the spiking activity of the fractional leaky integrate-and-fire model deviates from the spiking activity of the Markovian model and reflects the temporal accumulated intrinsic membrane dynamics that affect the response of the neuron to external stimulation.     

Combining particle tracking microrheology and viscometry for the study of DNA aqueous solutions

We use video particle tracking microrheology (VPTMR) in order to investigate the viscoelasticity of salmon DNA and correlate it to its steady-flow shear-thinning viscosity. Aqueous solutions of DNA are tested in a wide concentration range from the dilute to the semidilute unentangled concentration regime. The observed mean squared displacement shows power-law scaling with lag-time which is equivalent to power-law behavior of the complex modulus as a function of frequency that is, |G^*(ω)| = S ∙ ω ^α . The relaxation exponent α changes abruptly with concentration in the semidilute regime from about 1 to about 0.5 which is the exponent predicted by the Rouse model. The quasi-property S follows the scaling of viscosity for uncharged polymers near θ-conditions in the semidilute regime that is, with ν_eff = 0.50 − 0.51 . The shear-thinning exponent observed by viscometry increases gradually towards the value of 0.5 which has been predicted for Rouse chains under flow. Our findings are in agreement with recent studies of DNA solutions where DNA is treated as a model polymer and addresses the low-molar mass regime of DNA viscoelasticity. This work demonstrates that the combination of passive particle tracking with viscometry can provide a complete picture on the viscoelasticity of DNA-based biopolymer materials.     

The population genetic structure of clonal organisms generated by exponentially bounded and fat-tailed dispersal

Long-distance dispersal (LDD) plays an important role in many population processes like colonization, range expansion, and epidemics. LDD of small particles like fungal spores is often a result of turbulent wind dispersal and is best described by functions with power-law behavior in the tails ("fat tailed"). The influence of fat-tailed LDD on population genetic structure is reported in this article. In computer simulations, the population structure generated by power-law dispersal with exponents in the range of -2 to -1, in distinct contrast to that generated by exponential dispersal, has a fractal structure. As the power-law exponent becomes smaller, the distribution of individual genotypes becomes more self-similar at different scales. Common statistics like GST are not well suited to summarizing differences between the population genetic structures. Instead, fractal and self-similarity statistics demonstrated differences in structure arising from fat-tailed and exponential dispersal. When dispersal is fat tailed, a log-log plot of the Simpson index against distance between subpopulations has an approximately constant gradient over a large range of spatial scales. The fractal dimension D2 is linearly inversely related to the power-law exponent, with a slope of approximately -2. In a large simulation arena, fat-tailed LDD allows colonization of the entire space by all genotypes whereas exponentially bounded dispersal eventually confines all descendants of a single clonal lineage to a relatively small area.     

Minimizing errors in identifying Lévy flight behaviour of organisms

1. Lévy flights are specialized random walks with fundamental properties such as superdiffusivity and scale invariance that have recently been applied in optimal foraging theory. Lévy flights have movement lengths chosen from a probability distribution with a power-law tail, which theoretically increases the chances of a forager encountering new prey patches and may represent an optimal solution for foraging across complex, natural habitats. 2. An increasing number of studies are detecting Lévy behaviour in diverse organisms such as microbes, insects, birds, and mammals including humans. A principal method for detecting Lévy flight is whether the exponent (micro) of the power-law distribution of movement lengths falls within the range 1 < micro < or = 3. The exponent can be determined from the histogram of frequency vs. movement (step) lengths, but different plotting methods have been used to derive the Lévy exponent across different studies. 3. Here we investigate using simulations how different plotting methods influence the micro-value and show that the power-law plotting method based on 2(k) (logarithmic) binning with normalization prior to log transformation of both axes yields low error (1.4%) in identifying Lévy flights. Furthermore, increasing sample size reduced variation about the recovered values of micro, for example by 83% as sample number increased from n = 50 up to 5000. 4. Simple log transformation of the axes of the histogram of frequency vs. step length underestimated micro by c.40%, whereas two other methods, 2(k) (logarithmic) binning without normalization and calculation of a cumulative distribution function for the data, both estimate the regression slope as 1-micro. Correction of the slope therefore yields an accurate Lévy exponent with estimation errors of 1.4 and 4.5%, respectively. 5. Empirical reanalysis of data in published studies indicates that simple log transformation results in significant errors in estimating micro, which in turn affects reliability of the biological interpretation. The potential for detecting Lévy flight motion when it is not present is minimized by the approach described. We also show that using a large number of steps in movement analysis such as this will also increase the accuracy with which optimal Lévy flight behaviour can be detected.     

The effect of allometric partitioning on herbivory tolerance in four species in South China

Herbivory tolerance can offset the negative effects of herbivory on plants and plays an important role in both immigration and population establishment. Biomass reallocation is an important potential mechanism of herbivory tolerance. To understand how biomass allocation affects plant herbivory tolerance, it is necessary to distinguish the biomass allocations resulting from environmental gradients or plant growth. There is generally a tight balance between the amounts of biomass invested in different organs, which must be analyzed by means of an allometric model. The allometric exponent is not affected by individual growth and can reflect the changes in biomass allocation patterns of different parts. Therefore, the allometric exponent was chosen to study the relationship between biomass allocation pattern and herbivory tolerance. We selected four species (Wedelia chinensis, Wedelia trilobata, Merremia hederacea, and Mikania micrantha), two of which are invasive species and two of which are accompanying native species, and established three herbivory levels (0%, 25% and 50%) to compare differences in allometry. The biomass allocation in stems was negatively correlated with herbivory tolerance, while that in leaves was positively correlated with herbivory tolerance. Furthermore, the stability of the allometric exponent was related to tolerance, indicating that plants with the ability to maintain their biomass allocation patterns are more tolerant than those without this ability, and the tendency to allocate biomass to leaves rather than to stems or roots helps increase this tolerance. The allometric exponent was used to remove the effects of individual development on allocation pattern, allowing the relationship between biomass allocation and herbivory tolerance to be more accurately explored. This research used an allometric model to fit the nonlinear process of biomass partitioning during the growth and development of plants and provides a new understanding of the relationship between biomass allocation and herbivory tolerance.     

植物代谢速率与个体生物量关系研究进展

植物的各项生理生态功能（例如,呼吸、生长和繁殖）都与个体生物量成异速生长关系。West, Brown及Enquist基于分形网络结构理论所提出的WBE模型认为：植物的代谢（呼吸）速率正比于个体生物量的3/4次幂。然而,恒定的“3/4异速生长指数”与实测数据、植物生理生态学等研究之间存在矛盾,引发激烈的争论。论文分析了不同回归方法对代谢指数的影响,重点对植物代谢速率与个体生物量异速生长关系研究进展进行了综述,分析并得出了植物代谢指数在小个体时接近1.0,并随着生物量的增加而系统减小,且其密切依赖于氮含量的调控的结论。据此,提出了进一步深入研究植物代谢速率个体生物量关系需要解决的一些科学问题。     

The difference between above- and below-ground self-thinning lines in forest communities

Quantifying the self-thinning process in various plant communities has been a long-standing issue in both theoretical and empirical studies. Most studies on plant self-thinning have centered only on aboveground parts, and rarely on belowground parts. There is still a general lack of comparison between above- and belowground self-thinning processes, especially for forest communities. The fundamental mechanistic difference and the functional association between above- and belowground competition indicate that the self-thinning process of belowground parts may be different from that of aboveground parts. We investigated the self-thinning lines for above-ground (M _A), below-ground (M _B), and total biomass (M _T), respectively, across forest communities in China. The results showed that neither the classical self-thinning rule (−3/2 exponent) nor the universal scaling rule (−4/3 exponent) can apply to all the self-thinning relationships across these forest communities and that the self-thinning lines for belowground biomass were flatter and lower than those for aboveground biomass across most of these forest communities.