Statistical analyses support power law distributions found in neuronal avalanches

The size distribution of neuronal avalanches in cortical networks has been reported to follow a power law distribution with exponent close to -1.5, which is a reflection of long-range spatial correlations in spontaneous neuronal activity. However, identifying power law scaling in empirical data can be difficult and sometimes controversial. In the present study, we tested the power law hypothesis for neuronal avalanches by using more stringent statistical analyses. In particular, we performed the following steps: (i) analysis of finite-size scaling to identify scale-free dynamics in neuronal avalanches, (ii) model parameter estimation to determine the specific exponent of the power law, and (iii) comparison of the power law to alternative model distributions. Consistent with critical state dynamics, avalanche size distributions exhibited robust scaling behavior in which the maximum avalanche size was limited only by the spatial extent of sampling ("finite size" effect). This scale-free dynamics suggests the power law as a model for the distribution of avalanche sizes. Using both the Kolmogorov-Smirnov statistic and a maximum likelihood approach, we found the slope to be close to -1.5, which is in line with previous reports. Finally, the power law model for neuronal avalanches was compared to the exponential and to various heavy-tail distributions based on the Kolmogorov-Smirnov distance and by using a log-likelihood ratio test. Both the power law distribution without and with exponential cut-off provided significantly better fits to the cluster size distributions in neuronal avalanches than the exponential, the lognormal and the gamma distribution. In summary, our findings strongly support the power law scaling in neuronal avalanches, providing further evidence for critical state dynamics in superficial layers of cortex.     

Modeling bond correlations in denatured proteins and polypeptides

Bond‐orientational correlations for finite‐length homopolypeptides and a selected group of denatured proteins are obtained by numerical simulations using a polypeptide model with a potential of mean force. These correlations characterize the stiffness of the polypeptide backbone and are generally described by either an exponential or a power‐law decay in the asymptotic limit. However, for finite length polypeptides and unfolded proteins the correlations significantly deviate from either single exponential or power‐law behavior. A heuristic model is developed to analyze the correlations of homopolypeptides, which depends on the chain length and the side‐chain properties. The model contains power‐law and multi‐exponential terms, the latter which could be interpreted as local persistence lengths. In the asymptotic limit, the model reduces to the expected power‐law behavior. Simulations of denatured proteins show that the power‐law behavior of the correlations is significantly suppressed and only the multi‐exponential term is needed to model the correlations. In addition, average persistence lengths (ranging from 2.0 to 2.5 nm) are obtained from the correlations by fitting single exponentials and shown to be in general agreement with experiments, which also assume single exponential decay. © 2016 Wiley Periodicals, Inc. Biopolymers 105: 312–323, 2016.     

Spore Dispersal and Plant Disease Gradients; a Comparison between two Empirical Models

Power law and exponential models were fitted to 325 sets of observations which described decreases with distance in deposition of air-borne or splash-borne spores, or pollen, or in amounts of plant disease caused by fungi, bacteria or viruses. There, was generally little difference between the models in the goodness of fit to these data, although deposition gradients for spores borne in splash droplets were fitted better by exponential models and gradients for fungi with air-borne spores less than 10 μm in diameter were fitted better by power law models. The exponential model has the property that the observed variable decreases by half as the distance from the source increases by a constant increment (the half-distance); this provides a measureof the gradient that is more easy to visualize than the exponent in power law model. The half-distances of gradients for air-borne pathogens were greater than those for splash-borne or soil-borne pathogens. The exponential model is easier to incorporate into models of disease development than the power law model because the boundary condition at the source (the estimated number of spores or amount of disease at the source) is finite rather than infinite. However, both these empirical models have limitations and should not be extrapolated to distances outside the observed range.     

Scale dependency in the functional form of the distance decay relationship

We examine a novel mathematical approach which posits that the decay of similarity in community composition with increasing distance (aka distance decay) can be modeled as the sum of individual species joint‐probability vs distance relationships. Our model, supported by analyses of these curves from three datasets (North American breeding birds, North American taiga plants, and tropical forest trees), suggest that when sampling grain is large enough to avoid absences due to stochastic sampling effects, and/or sampling extent is large enough to generate species turnover through the deterministic crossing of environmental and/or geographical range limits, species joint‐probability over increasing distance will generally exhibit exponential decay. However, at small scales where occurrence is driven more by stochastic sampling effects, species joint‐probability curves exhibit a power‐law decay form. Lacking a theoretical prediction of how individual species joint‐probability relationships combine to generate community distance decay, we also performed a meta‐analysis of 26 ecological and 4 human‐system datasets, using non‐linear regression to mean and quantile non‐linear regression at tau = 0.95 for linear, exponential, and power‐law decay forms. These analyses demonstrate that the functional form of community distance decay – as shown by comparison of AIC ranks – is largely determined by observational scale, with power law decay prevailing within domains where the species pool remains constant, while exponential decay prevails at larger scales over which the species pool varies, paralleling the patterns predicted in our mathematical approach.     

Distributions of the use frequencies of amino acids in the hypervariable regions of immunoglobulins

Two types of distributions for the frequencies of occurrence of amino acids in each position of hypervariable regions CDR-1 and CDR-2 were obtained for 2,000 immunoglobulins. The results show that some positions fit an inverse power-law distribution, while others fit an exponential-type distribution. As a result of comparison with structural data in the literature it is proposed that sites in which the frequency distribution fits the inverse power law are critical to maintaining canonical shapes of the recognition regions or are involved in modulating these canonical conformations, while those sites where the distribution fits the exponential law are those which should be exclusively involved in the recognition mechanism. Correspondence to: F. Lara-Ochoa     

Power law versus exponential state transition dynamics: application to sleep-wake architecture

Background

Despite the common experience that interrupted sleep has a negative impact on waking function, the features of human sleep-wake architecture that best distinguish sleep continuity versus fragmentation remain elusive. In this regard, there is growing interest in characterizing sleep architecture using models of the temporal dynamics of sleep-wake stage transitions. In humans and other mammals, the state transitions defining sleep and wake bout durations have been described with exponential and power law models, respectively. However, sleep-wake stage distributions are often complex, and distinguishing between exponential and power law processes is not always straightforward. Although mono-exponential distributions are distinct from power law distributions, multi-exponential distributions may in fact resemble power laws by appearing linear on a log-log plot.

Methodology/Principal Findings

To characterize the parameters that may allow these distributions to mimic one another, we systematically fitted multi-exponential-generated distributions with a power law model, and power law-generated distributions with multi-exponential models. We used the Kolmogorov-Smirnov method to investigate goodness of fit for the “incorrect” model over a range of parameters. The “zone of mimicry” of parameters that increased the risk of mistakenly accepting power law fitting resembled empiric time constants obtained in human sleep and wake bout distributions.

Conclusions/Significance

Recognizing this uncertainty in model distinction impacts interpretation of transition dynamics (self-organizing versus probabilistic), and the generation of predictive models for clinical classification of normal and pathological sleep architecture.     

北京东灵山土壤动物多样性海拔格局的尺度推绎规律

识别群落内部各类群多样性格局的复杂性是生态学家面临的挑战,而尺度推绎规律是揭示复杂生态结构的有效途径之一。本研究利用多重分形的方法探索了不同海拔土壤动物多样性格局的尺度推绎规律,对比分析了凋落物层和土壤层之间多重分形谱的差异。结果表明: 与之前对植物群落的分析结果相似,土壤动物多样性尺度推绎规律同样具有幂律特征,如丰富度、Shannon多样性指数和Simpson多样性的倒数。凋落物层和土壤层中不同相对多度土壤动物的丰富度也具有幂律尺度推绎规律。凋落物层和土壤层中土壤动物多样性格局都具有多重分形特征,但凋落物层中多样性的分形结构比土壤层更均匀,且两层间优势类群与稀有类群的尺度推绎特征在多重分形谱上不同格局。幂律尺度推绎规律对于有着较高丰富度与多度的土壤动物同样存在,从而有助于揭示地下生物多样性的空间分布机制。     

Characterising the micro-mechanical behaviour of the carious dentine of primary teeth using nano-indentation

A better appreciation of the properties of carious dentine would be of clinical advantage in carious assessment and management. The aim of this study is to understand the deterioration of the mechanical properties of carious dentine as a result of bacterial demineralising process as well as change in dentine structures observed under scanning electronic microscope. Eight primary molar teeth with untreated carious dentine were axially sectioned and fine polished for nano-indentation. On each specimen, six lines of indentation, evenly distributed through the lesion, were made from the pulp to lesion cavity floor parallel to tubule direction using nano-indentation (Ultra Micro Indentation System, UMIS-2000), while another two indentation lines were made on an adjacent region of sound dentine in the same manner. All tests were conducted on hydrated specimens. Hardness and elastic modulus decreased significantly and progressively toward the cavity floor varying from 0.56 to 0.001 GPa and 14.55 to 0.015 GPa, respectively. The change in mechanical properties was in a specific pattern as a function of lesion depth, in which the hardness could be fitted to an exponential function, while the variation of the elastic modulus across the entire lesion was fitted to a power law relationship. More critical evaluation of the elastic modulus data indicated that two distinct exponential functions provided an excellent fit to the results. These changes in elastic modulus also matched the structural changes seen across a lesion, which were associated with a change from primarily peritubular to intertubular dissolution.     

Quaternary relaxations in sol-gel encapsulated hemoglobin studied via NIR and UV spectroscopy

In this work, we study the kinetics of the R --> T transition in hemoglobin using a combination of near-infrared and near-ultraviolet spectroscopy. We use a sol-gel encapsulation protocol to decelerate the conformational transitions and to avoid spectral perturbations arising from ligand migration and recombination. We monitor two spectroscopic markers: band III in the near-IR, which is a fine probe of the heme pocket conformation, and the tryptophan band in the near-UV, which probes the formation of the Trpbeta37-Aspalpha94 hydrogen bond, characteristic of the T structure, at the critical alpha1beta2 subunit interface. The time evolution of these two bands is monitored after deoxygenation of encapsulated oxyhemoglobin, obtained by diffusion of a reducing agent into the porous silica matrix. Characteristic spectral shifts are observed: comparison with myoglobin enables us to assign them to quaternary structure relaxations. Band III spectral relaxation is clearly nonexponential, and analysis with the Maximum Entropy Method enables us to identify three processes. On the other hand, near-UV spectral relaxation follows an exponential decay with a time constant closely corresponding to the second process observed in the near IR. Very interestingly, the rates of all processes markedly depend on the viscosity of the co-encapsulated solvent, following a power law. Our results reveal correlations between heme pocket relaxations, induced by the R --> T transition, and structural event(s) occurring at the alpha1beta2 interface and highlight their solvent dependence. The power law viscosity dependence of relaxation rates suggests that the observed protein relaxations are "slaved" to the co-encapsulated solvent. The stepwise character of the quaternary transition is also evidenced.     

Power Law Distribution Defines Structural Disorder as a Structural Element Directly Linked with Function

Although intrinsically disordered proteins are prevalent and functionally important, it has never been asked whether structural disorder should be considered as a separate structural category on its own or merely as a lack of secondary and/or tertiary structure. We address this issue by showing that its length distribution in the human proteome follows a power law, with many short regions but also a significant incidence of very long disordered regions. This behavior is in sharp contrast with that of conventional secondary structural elements and is highly reminiscent of the distribution of tertiary structural units in proteins. We interpret this finding by the direct functional involvement of disorder, which distinguishes it from secondary structural elements and endows it with tertiary structural attributes.     

Changes in the salivary cotinine cut-offs to discriminate smokers and non-smokers before and after Spanish smoke-free legislation

IntroductionHigh levels of cotinine in non-smokers indicate passive exposure to tobacco smoke. This study aims to evaluate variations in salivary cotinine cut-offs to discriminate smokers and non-smokers before and after the implementation of smoke-free legislation (Law 28/2005 and Law 42/2010) in a sample of the adult population of Barcelona, Spain.MethodsThis longitudinal study analyzes salivary cotinine samples and self-reported information from a representative sample (n = 676) of the adult population from Barcelona before and after the approval of smoke-free legislation. We calculated the receiver operating characteristic (ROC) curves, to obtain optimal cotinine cut-off points to discriminate between smokers and non-smokers overall, by sex and age, and their corresponding sensitivity, specificity, and area under the curve. We used linear mixed-effects models, with individuals as random effects, to model the percentage change of cotinine concentration before and after the implementation of both laws.ResultsThe mean salivary cotinine concentration was significantly lower post-2010 law (−85.8%, p < 0.001). The ROC curves determined that the optimal cotinine cut-off points for discriminating non-smokers and smokers were 10.8 ng/mL (pre-2005 law) and 5.6 ng/mL (post-2010 law), with a post-2010 law sensitivity of 92.6%, specificity of 98.4%, and an area under the curve of 97.0%. The post-2010 law cotinine cut-off points were 5.6 ng/mL for males and 1.9 ng/mL for females.ConclusionThe implementation of Spanish smoke-free legislation was effective in reducing secondhand smoke exposure and, therefore, also in reducing the cut-off point for salivary cotinine concentration. This value should be used to better assess tobacco smoke exposure in this population.     

Uncovering Patterns of Inter-Urban Trip and Spatial Interaction from Social Media Check-In Data

The article revisits spatial interaction and distance decay from the perspective of human mobility patterns and spatially-embedded networks based on an empirical data set. We extract nationwide inter-urban movements in China from a check-in data set that covers half a million individuals within 370 cities to analyze the underlying patterns of trips and spatial interactions. By fitting the gravity model, we find that the observed spatial interactions are governed by a power law distance decay effect. The obtained gravity model also closely reproduces the exponential trip displacement distribution. The movement of an individual, however, may not obey the same distance decay effect, leading to an ecological fallacy. We also construct a spatial network where the edge weights denote the interaction strengths. The communities detected from the network are spatially cohesive and roughly consistent with province boundaries. We attribute this pattern to different distance decay parameters between intra-province and inter-province trips.     

Power spectra of extinction in the fossil record

Recent Fourier analyses of fossil extinction data have indicated that the power spectrum of extinction during the Phanerozoic may take the form of 1/f noise, a result which, it has been suggested, could be indicative of the presence of `critical dynamics'' in the processes giving rise to extinction. In this paper we examine extinction power spectra in some detail, using family-level data from two widely available compilations. We find that although the average form of the power spectrum roughly obeys the 1/f law, the spectrum can be represented more accurately by dividing it into two regimes: a low-frequency one which is well fit by an exponential, and a high-frequency one in which it follows a power law with a 1/f² form. We give explanations for the occurrence of each of these behaviours and for the position of the crossover between them.     

Growth of Ca-D-malate crystals in a bioreactor

To develop a bioreactor for solid-to-solid conversions, the conversion of solid Ca-maleate to solid Ca-D-malate by permeabilized Pseudomonas pseudoalcaligenes was studied. In a bioreactor seeded with product (Ca-D-malate) crystals, growth of Ca-D-malate crystals is the last step in the solid-to-solid conversion and is described here. Crystal growth is described as a transport process followed by surface processes. In contrast to the linear rate law obeyed by the transport process, the surface processes of a crystal-growth process can also obey a parabolic or exponential rate law. Growth of Ca-D-malate crystals from a supersaturated aqueous solution was found to be surface-controlled and obeyed an exponential rate law. Based on this rate law, a kinetic model was developed which describes the decrease in supersaturation due to Ca-D-malate crystal growth as a function of the constituent ions, Ca(2+) and D-malate(2-). The kinetic parameters depended on temperature, but, as expected (surface-controlled), they were hardly affected by the stirring speed.     

Does the disturbance hypothesis explain the biomass increase in basin‐wide Amazon forest plot data?

Positive aboveground biomass trends have been reported from old-growth forests across the Amazon basin and hypothesized to reflect a large-scale response to exterior forcing. The result could, however, be an artefact due to a sampling bias induced by the nature of forest growth dynamics. Here, we characterize statistically the disturbance process in Amazon old-growth forests as recorded in 135 forest plots of the RAINFOR network up to 2006, and other independent research programmes, and explore the consequences of sampling artefacts using a data-based stochastic simulator. Over the observed range of annual aboveground biomass losses, standard statistical tests show that the distribution of biomass losses through mortality follow an exponential or near-identical Weibull probability distribution and not a power law as assumed by others. The simulator was parameterized using both an exponential disturbance probability distribution as well as a mixed exponential–power law distribution to account for potential large-scale blowdown events. In both cases, sampling biases turn out to be too small to explain the gains detected by the extended RAINFOR plot network. This result lends further support to the notion that currently observed biomass gains for intact forests across the Amazon are actually occurring over large scales at the current time, presumably as a response to climate change.     

Process‐based functions for seed retention on animals: a test of improved descriptions of dispersal using multiple data sets

Studies of external seed transport on animals usually assume that the probability of detachment is constant, so that seed retention should show a simple exponential relationship with time. This assumption has not been tested explicitly, and may lead to inaccurate representation of long distance seed dispersal by animals. We test the assumption by comparing the fit to empirical data of simple, two‐parameter functions. Fifty‐two data sets were obtained from five published studies, describing seed retention of 32 plant species on sheep, cattle, deer, goats and mice. Model selection suggested a simple exponential function was adequate for data sets in which seed retention was followed for short periods ( <48 h). The data gathered over longer periods (49–219 days) were best described by the power exponential function, a form of the stretched exponential which allows a changing dropping rate. In these cases the power exponential showed that seed dropping rate decreased with time, suggesting that seeds vary in attachment, with some seeds becoming deeply buried or wound up in the animal's coat. Comparison of fitted parameters across all the data sets also confirmed that seeds with adhesive structures have lower dropping rates than those without. We conclude that the seed dropping rate often changes with time during external transport on animals and that the power exponential is an effective function to describe this change. We advise that, to analyse seed dropping rates adequately, retention should be measured over reasonable time periods – until most seeds are dropped – and both the simple and power exponential functions should be fitted to the resulting data. To increase its utility, we provide functions describing the seed dropping rate and the dispersal kernel resulting from the power exponential relationship.     

The Universal Statistical Distributions of the Affinity,Equilibrium Constants,Kinetics and Specificity in Biomolecular Recognition

We uncovered the universal statistical laws for the biomolecular recognition/binding process. We quantified the statistical energy landscapes for binding, from which we can characterize the distributions of the binding free energy (affinity), the equilibrium constants, the kinetics and the specificity by exploring the different ligands binding with a particular receptor. The results of the analytical studies are confirmed by the microscopic flexible docking simulations. The distribution of binding affinity is Gaussian around the mean and becomes exponential near the tail. The equilibrium constants of the binding follow a log-normal distribution around the mean and a power law distribution in the tail. The intrinsic specificity for biomolecular recognition measures the degree of discrimination of native versus non-native binding and the optimization of which becomes the maximization of the ratio of the free energy gap between the native state and the average of non-native states versus the roughness measured by the variance of the free energy landscape around its mean. The intrinsic specificity obeys a Gaussian distribution near the mean and an exponential distribution near the tail. Furthermore, the kinetics of binding follows a log-normal distribution near the mean and a power law distribution at the tail. Our study provides new insights into the statistical nature of thermodynamics, kinetics and function from different ligands binding with a specific receptor or equivalently specific ligand binding with different receptors. The elucidation of distributions of the kinetics and free energy has guiding roles in studying biomolecular recognition and function through small-molecule evolution and chemical genetics.     

Exponential growth and allometric equations

Earlier the simple exponential growth law has been used when attempts have been made to explain theoretically the existence of allometric equations in biology (Bertalanffy, 1968; Giinther, 1975). We show here that the exponential growth law in which the proportion coefficient also depends on time is a better approximation. We consider especially the metabolic rate and the body mass of man during the growth process.     

A temporal model of animal behavior based on a fractality in the feeding of Drosophila melanogaster

We present a new temporal model of animal behavior based on the ethological idea that the internal states of the individual essentially determine the behavior. The internal states, however, are conditioned by the external stimuli. This model, including environmental and internal parameters, predicts a fractal property of the behavior, that is, an inverse power law distribution of the duration. Being consistent with the model, we have found a fractal property of feeding in Drosophila melanogaster: The dwelling time of starved flies on food showed a clear inverse power law distribution. The dependence of the fractal dimension on the intensity of food stimuli has been observed, and the predicted change into an exponential distribution was proved.