An assessment of preferential attachment as a mechanism for human sexual network formation

Recent research into the properties of human sexual-contact networks has suggested that the degree distribution of the contact graph exhibits power-law scaling. One notable property of this power-law scaling is that the epidemic threshold for the population disappears when the scaling exponent rho is in the range 2 < rho < or = 3. This property is of fundamental significance for the control of sexually transmitted diseases (STDs) such as HIV/AIDS since it implies that an STD can persist regardless of its transmissibility. A stochastic process, known as preferential attachment, that yields one form of power-law scaling has been suggested to underlie the scaling of sexual degree distributions. The limiting distribution of this preferential attachment process is the Yule distribution, which we fit using maximum likelihood to local network data from samples of three populations: (i) the Rakai district, Uganda; (ii) Sweden; and (iii) the USA. For all local networks but one, our interval estimates of the scaling parameters are in the range where epidemic thresholds exist. The estimate of the exponent for male networks in the USA is close to 3, but the preferential attachment model is a very poor fit to these data. We conclude that the epidemic thresholds implied by this model exist in both single-sex and two-sex epidemic model formulations. A strong conclusion that we derive from these results is that public health interventions aimed at reducing the transmissibility of STD pathogens, such as implementing condom use or high-activity anti-retroviral therapy, have the potential to bring a population below the epidemic transition, even in populations exhibiting large degrees of behavioural heterogeneity.     

Is non‐loglinear allometry a statistical artifact?

The allometric equation, y = ax^b, is commonly fitted to data indirectly by transforming predictor (x) and response (y) variables to logarithms, fitting a straight line to the transformations, and then back‐transforming (exponentiating) the resulting equation to the original arithmetic scale. Sometimes, however, transformation fails to linearize the observations, thereby giving rise to what has come to be known as non‐loglinear allometry. A smooth curve for observations displayed on a log–log plot is usually interpreted to mean that the scaling exponent in the allometric equation is a continuously changing function of body size, whereas a breakpoint between two (or more) linear segments on a log–log plot is typically taken to mean that the exponent changes abruptly, coincident with some important milestone in development. I applied simple graphical and statistical procedures in re‐analyses of three well‐known examples of non‐loglinear allometry, and showed in every instance that the relationship between predictor and response can be described in the original scale by simple functions with constant values for the exponent b. In no instance does the allometric exponent change during the course of development. Transformation of data to logarithms created new distributions that actually obscured the relationships between predictor and response variables in these investigations, and led to erroneous perceptions of growth. Such confounding effects of transformation are not limited to non‐loglinear allometry but are common to all applications of the allometric method. © 2012 The Linnean Society of London, Biological Journal of the Linnean Society, 2012, ?? , ??–??.     

Model selection and logarithmic transformation in allometric analysis

The standard approach to most allometric research is to gather data on a biological function and a measure of body size, convert the data to logarithms, display the new values in a bivariate plot, and then fit a straight line to the transformations by the method of least squares. The slope of the fitted line provides an estimate for the allometric (or scaling) exponent, which often is interpreted in the context of underlying principles of structural and functional design. However, interpretations of this sort are based on the implicit assumption that the original data conform with a power function having an intercept of 0 on a plot with arithmetic coordinates. Whenever this assumption is not satisfied, the resulting estimate for the allometric exponent may be seriously biased and misleading. The problem of identifying an appropriate function is compounded by the logarithmic transformations, which alter the relationship between the original variables and frequently conceal the presence of outliers having an undue influence on properties of the fitted equation, including the estimate for the allometric exponent. Much of the current controversy in allometric research probably can be traced to substantive biases introduced by investigators who followed standard practice. We illustrate such biases with examples taken from the literature and outline a general methodology by which the biases can be minimized in future research.     

Extended Fujita approach to the molecular weight distribution of polysaccharides and other polymeric systems

In 1962 H. Fujita (H. Fujita, Mathematical Theory of Sedimentation Analysis, Academic Press, New York, 1962) examined the possibility of transforming a quasi-continuous distribution g(s) of sedimentation coefficient s into a distribution f(M) of molecular weight M for linear polymers using the relation f(M)=g(s)·(ds/dM) and showed that this could be done if information about the relation between s and M is available from other sources. Fujita provided the transformation based on the scaling relation s=κ(s)M(0.5), where κ(s) is taken as a constant for that particular polymer and the exponent 0.5 essentially corresponds to a randomly coiled polymer under ideal conditions. This method has been successfully applied to mucus glycoproteins (S.E. Harding, Adv. Carbohyd. Chem. Biochem. 47 (1989) 345-381). We now describe an extension of the method to general conformation types via the scaling relation s=κM(b), where b=0.4-0.5 for a coil, ～0.15-0.2 for a rod and ～0.67 for a sphere. We give examples of distributions f(M) versus M obtained for polysaccharides from SEDFIT derived least squares g(s) versus s profiles (P. Schuck, Biophys. J. 78 (2000) 1606-1619) and the analytical derivative for ds/dM performed with Microcal ORIGIN. We also describe a more direct route from a direct numerical solution of the integral equation describing the molecular weight distribution problem. Both routes give identical distributions although the latter offers the advantage of being incorporated completely within SEDFIT. The method currently assumes that solutions behave ideally: sedimentation velocity has the major advantage over sedimentation equilibrium in that concentrations less than 0.2mg/ml can be employed, and for many systems non-ideality effects can be reasonably ignored. For large, non-globular polymer systems, diffusive contributions are also likely to be small.     

N-(2-Carboxyethyl)chitosans: regioselective synthesis,characterisation and protolytic equilibria

N-(2-Carboxyethyl)chitosans were obtained by reaction of low molecular weight chitosan with a low degree of acetylation and 3-halopropionic acids under mild alkaline media (pH 8-9, NaHCO3) at 60 degrees C. The chemical structure of the derivatives obtained was determined by 1H and 13C NMR spectroscopies. It was found that alkylation of chitosan by 3-halopropionic acids proceeds exclusively at the amino groups. The products obtained are described in terms of their degrees of carboxyethylation and ratio of mono-, di-substitution and free amine content. The protonation constants of amino and carboxylate groups of a series of N-(2-carboxyethyl)chitosans were determined by pH-titration at ionic strength 0.1 M KNO3 and 25 degrees C.     

Denatured state thermodynamics: residual structure, chain stiffness and scaling factors

A set of nine variants of yeast iso-1-cytochrome c with zero or one surface histidine have been engineered such that the N-terminal amino group is acetylated in vivo. N-terminal acetylation has been confirmed by mass spectral analysis of intact and proteolytically digested protein. The histidine-heme loop-forming equilibrium, under denaturing conditions (3 M guanidine hydrochloride), has been measured by pH titration providing an observed pK(a), pK(a)(obs), for each variant. N-terminal acetylation prevents the N-terminal amino group-heme binding equilibrium from interfering with measurements of histidine-heme affinity. Significant deviation is observed from the linear dependence of pK(a)(obs) on the log of the number of monomers in the loop formed, expected for a random coil denatured state. The maximum histidine-heme affinity occurs for a loop size of 37 monomers. For loop sizes of 37-83 monomers, histidine-heme pK(a)(obs) values are consistent with a scaling factor of -4.2+/-0.3. This value is much larger than the scaling factor of -1.5 for a freely jointed random coil, which is commonly used to represent the conformational properties of protein denatured states. For loop sizes of nine to 22 monomers, chain stiffness is likely responsible for the decreases in histidine-heme affinity relative to a loop size of 37. The results are discussed in terms of residual structure and sequence composition effects on the conformational properties of the denatured states of proteins.     

Studies on the antibacterial activities and mechanisms of chitosan obtained from cuticles of housefly larvae

Chitosan was obtained from cuticles of the housefly (Musca domestica) larvae. Antibacterial activities of different Mw chitosans were examined against six bacteria. Antibacterial mechanisms of chitosan were investigated by measuring permeability of bacterial cell membranes and observing integrity of bacterial cells. Results show that the antibacterial activity of chitosan decreased with increase in Mw. Chitosan showed higher antibacterial activity at low pH. Ca2+ and Mg2+ could markedly reduce the antibacterial activity of chitosan. The minimum inhibitory concentrations of chitosans ranged from 0.03% - 0.25% and varied with the type of bacteria and Mw of chitosan. Chitosan could cause leakage of cell contents of the bacteria and disrupt the cell wall.     

Temperature dependence of collagen fluorescence.

Dermal collagens have several fluorescent moieties in the UV and visible spectral regions that may serve as molecular probes of collagen. We studied the temperature-dependence of a commercial calf skin collagen and acid-extracted Skh-1 hairless mouse collagen at temperatures from 9 degrees C to 60 degrees C for excitation/emission wavelengths 270/305 nm (tyrosine), 270/360 nm (excimer-like aggregated species), 325/400 nm (dityrosine) and 370/450 nm (glycation adduct). L-tyrosine (1 x 10(-5) M in 0.5 M HOAc) acted as a "reference compound" devoid of any collagen structural effects. In general, the fluorescence efficiency of these fluorophores decreases with increasing temperature. Assuming that rate constant for fluorescence deactivation has the form k(d)(T) = k(d) degrees exp (-DeltaE/RT), an Arrhenius plot of log[(1/Phi) - 1] vs. 1/T affords a straight line whose (negative) slope is proportional to the activation energy, DeltaE, of the radiationless process(es) that compete with fluorescence. Because it is difficult to accurately measure Phi(f) for collagen-bound fluorophores, we derived an approximate formula for an activation parameter, DeltaE*, evaluated from an Arrhenius-like plot of log 1/I(N)vs. 1/T, (1/I(N)vs. is the reciprocal normalized fluorescence intensity). Tyrosine in dilute solution affords a linear Arrhenius plot in both of the above cases. Using the known value of Phi(f) = 0.21 for free tyrosine at room temperature, we determined that DeltaE* is accurate to approximately 25% in the present instance. Collagen curves are non-linear, but they are quasi-linear below approximately 20 degrees C, where the helical form predominates. Values of DeltaE* determined from the data at T < 20 degrees C ranged from 6.2-8.4 kJ mol(-1) (1.5-2.0 kcal mol(-1)) for mouse collagen and 10.3-11.4 kJ mol(-1) (2.5-2.7 kcal mol(-1)) for calf skin collagen, consistent with collisional deactivation of the fluorescent state via thermally enhanced molecular vibrations and rotations. Above 20 degrees C, log 1/I(N)vs. 1/T plots from Skh-1 hairless mouse collagen are concave-downward, suggesting that fluorescence deactivation from the denatured coil has a significant temperature-independent component. For calf skin collagen, these plots are concave-upward, suggesting an increase in activation energy above Tm. These results suggest that collagen backbone and supramolecular structure can influence the temperature dependence of the bound fluorophores, indicating the future possibility of using activation data as a probe of supramolecular structure and conformation.     

The observed range for temporal mean-variance scaling exponents can be explained by reproductive correlation

The mean-variance scaling relationship known as Taylor's power law has been well documented empirically over the past four decades but a general theoretical explanation for the phenomenon does not exist. Here we provide an explanation that relates empirical patterns of temporal mean-variance scaling to individual level reproductive behavior. Initially, we review the scaling behavior of population growth models to establish theoretical limits for the scaling exponent b that is in agreement with the empirically observed range (1≤b≤2). We go on to show that the degree of reproductive covariance among individuals determines the scaling exponent b. Independent reproduction results in an exponent of one, while completely correlated reproduction results in the upper limit of two. Intermediate exponents, which are common empirically, can be generated through the decay of reproductive covariance with increasing population size. Finally, we describe how the link between reproductive correlation and the scaling exponent provides a way to infer properties of individual-level reproductive behavior, such as the relative influence of demographic stochasticity, from a macroecological pattern.     

Viscometric and static light scattering studies on an exopolysaccharide produced by Lactobacillus delbrueckii subspecies bulgaricus NCFB 2483

The rheological properties and molecular parameters of a purified exopolysaccharide (EPS) produced by a ropy strain of Lactobacillus delbrueckii subsp. bulgaricus NCFB 2483 were investigated. Using capillary viscometry, an intrinsic viscosity of 2,013 mL/g was obtained. The flow curves were fitted by both the Carreau and the Cross equations for shear-thinning fluids, with the Carreau equation giving a better fit. The Cross equation fitted fairly well the plot of reduced viscosity as a function of reduced shear rate with an exponent value (1 - n) of approximately 0.76, typical of random coil polymers. Furthermore, the concentration dependence of the viscosity plot showed a gradient of approximately 1.1 in the dilute regime and 3.3 in the semidilute regime. Molecular parameters were obtained using a multiangle laser light scattering technique. The 2483 EPS molecules had a weight-average molar mass of approximately 2 x 10(6) Da and a z-average root mean square radius (RMS) of approximately 151 nm. From the light scattering data, the bacterial EPS was also found to have a low polydispersity index (approximately 1.15) and adopt a random coil conformation.     

Discovery of Power-Law Growth in the Self-Renewal of Heterogeneous Glioma Stem Cell Populations

Background

Accumulating evidence indicates that cancer stem cells (CSCs) drive tumorigenesis. This suggests that CSCs should make ideal therapeutic targets. However, because CSC populations in tumors appear heterogeneous, it remains unclear how CSCs might be effectively targeted. To investigate the mechanisms by which CSC populations maintain heterogeneity during self-renewal, we established a glioma sphere (GS) forming model, to generate a population in which glioma stem cells (GSCs) become enriched. We hypothesized, based on the clonal evolution concept, that with each passage in culture, heterogeneous clonal sublines of GSs are generated that progressively show increased proliferative ability.

Methodology/Principal Findings

To test this hypothesis, we determined whether, with each passage, glioma neurosphere culture generated from four different glioma cell lines become progressively proliferative (i.e., enriched in large spheres). Rather than monitoring self-renewal, we measured heterogeneity based on neurosphere clone sizes (#cells/clone). Log-log plots of distributions of clone sizes yielded a good fit (r>0.90) to a straight line (log(% total clones) = k*log(#cells/clone)) indicating that the system follows a power-law (y = x^k) with a specific degree exponent (k = −1.42). Repeated passaging of the total GS population showed that the same power-law was maintained over six passages (CV = −1.01 to −1.17). Surprisingly, passage of either isolated small or large subclones generated fully heterogeneous populations that retained the original power-law-dependent heterogeneity. The anti-GSC agent Temozolomide, which is well known as a standard therapy for glioblastoma multiforme (GBM), suppressed the self-renewal of clones, but it never disrupted the power-law behavior of a GS population.

Conclusions/Significance

Although the data above did not support the stated hypothesis, they did strongly suggest a novel mechanism that underlies CSC heterogeneity. They indicate that power-law growth governs the self-renewal of heterogeneous glioma stem cell populations. That the data always fit a power-law suggests that: (i) clone sizes follow continuous, non-random, and scale-free hierarchy; (ii) precise biologic rules that reflect self-organizing emergent behaviors govern the generation of neurospheres. That the power-law behavior and the original GS heterogeneity are maintained over multiple passages indicates that these rules are invariant. These self-organizing mechanisms very likely underlie tumor heterogeneity during tumor growth. Discovery of this power-law behavior provides a mechanism that could be targeted in the development of new, more effective, anti-cancer agents.     

Degradation of chitosans with chitinase B from Serratia marcescens. Production of chito-oligosaccharides and insight into enzyme processivity

Family 18 chitinases such as chitinase B (ChiB) from Serratia marcescens catalyze glycoside hydrolysis via a mechanism involving the N-acetyl group of the sugar bound to the -1 subsite. We have studied the degradation of the soluble heteropolymer chitosan, to obtain further insight into catalysis in ChiB and to experimentally assess the proposed processive action of this enzyme. Degradation of chitosans with varying degrees of acetylation was monitored by following the size-distribution of oligomers, and oligomers were isolated and partly sequenced using (1)H-NMR spectroscopy. Degradation of a chitosan with 65% acetylated units showed that ChiB is an exo-enzyme which degrades the polymer chains from their nonreducing ends. The degradation showed biphasic kinetics: the faster phase is dominated by cleavage on the reducing side of two acetylated units (occupying subsites -2 and -1), while the slower kinetic phase reflects cleavage on the reducing side of a deacetylated and an acetylated unit (bound to subsites -2 and -1, respectively). The enzyme did not show preferences with respect to acetylation of the sugar bound in the +1 subsite. Thus, the preference for an acetylated unit is absolute in the -1 subsite, whereas substrate specificity is less stringent in the -2 and +1 subsites. Consequently, even chitosans with low degrees of acetylation could be degraded by ChiB, permitting the production of mixtures of oligosaccharides with different size distributions and chemical composition. Initially, the degradation of the 65% acetylated chitosan almost exclusively yielded oligomers with even-numbered chain lengths. This provides experimental evidence for a processive mode of action, moving the sugar chain two residues at a time. The results show that nonproductive binding events are not necessarily followed by substrate release but rather by consecutive relocations of the sugar chain.     

Physicochemical properties and antitumor activities of water-soluble native and sulfated hyperbranched mushroom polysaccharides

A water-soluble hyperbranched beta-glucan, coded as TM3b, extracted from sclerotia of an edible fungus (Pleurotus tuber-regium) was fractioned into eight fractions coded as F1-F8 by a nonsolvent addition method. Five fractions were treated with chlorosulfonic acid at 35 degrees C to synthesize successfully sulfated derivatives coded as S-F2, S-F3, S-F4, S-F5, and S-F8 with degree of substitution of 0.28-0.54. The 13C NMR results of these sulfated beta-glucans indicated that while the C-6 position was fully substituted, C-2, C-3, and C-4 were only partially substituted by the sulfate groups. The weight-average molecular weights (Mw) and intrinsic viscosities ([eta]) of the native and sulfated TM3b fractions were determined using multi-angle laser light scattering and viscometry in 0.15M aq NaCl at 25 degrees C, respectively. The dependences of [eta] on Mw for TM3b and sulfated TM3b were found to be [eta]=0.18Mw(0.28+/-0.03) (Mw range from 3.30 x 10(4) to 3.90 x 10(7)) and [eta]=2.24 x 10(-2)Mw(0.52+/-0.06) (Mw range from 3.24 x 10(4) to 3.15 x 10(5)) in 0.15M aq NaCl at 25 degrees C, respectively. It revealed that both the native TM3b and its sulfated derivatives exist in a spherical chain conformation in 0.15M aq NaCl. Furthermore, the native and sulfated TM3b fractions showed potent antitumor activities in vivo and in vitro. The sulfated derivatives exhibited relatively higher in vitro antitumor activity against human hepatic cancer cell line HepG2 than the native TM3b. Water solubility and introduction of sulfate groups were the main factors in enhancing the antitumor activities.     

Inactivation of bacillus spores in dry systems at low and high temperatures.

A plot of the thermal resistance of Bacillus subtilis var. niger spores (log D value) against temperature was linear between 37 and 190 degrees C (z = 23 degrees C), provided that the relative humidity of the spore environment was kept below a certain critical level. The corresponding plot for Bacillus stearothermophilus spores was linear in the range 150 to 180 degrees C (z = 29 degrees C) but departed from linearity at lower temperatures (decreasing z value). However, the z value of 29 degrees C was decreased to 23 degrees C if spores were dried before heat treatment. The straight line corresponding to this new z value was consistent with the inactivation rate at a lower temperature (60 degrees C). The data indicate that bacterial spores which are treated in dry heat at an environmental relative humidity near zero are inactivated mainly by a drying process. By extrapolation of the thermal resistance plot obtained under these conditions for B. subtilis var. niger spores, the D value at 0 degrees C would be about 4 years.     

N-(carboxymethylidene)chitosans and N-(carboxymethyl)chitosans: Novel chelating polyampholytes obtained from chitosan glyoxylate

Glyoxylic acid, added to aqueous suspensions of chitosan, causes immediate dissolution of chitosan and gel formation within 3–4 h if the pH is 4.5–5.5. Solutions at lower pH values gel after 2 min of warming at 60–80°. Chitosan glyoxylate solutions brought to alkaline pH with sodium hydroxide do not precipitate chitosan. Evidence is given that a Schiff base, namely N-(carboxymethylidene)chitosan, is formed. N-(Carboxymethylidene)chitosans are reduced by sodium cyanoborohydride at room temperature to give N-(carboxymethyl)chitosans, obtained as white, free-flowing powders, soluble in water at all pH values. A series of N-(carboxymethyl)chitosans having various degrees of acetylation and N-carboxymethylation was obtained, and characterized by viscometry, elemental analysis, and i.r. spectrometry. For the fully substituted N-(carboxymethyl)chitosans, the pK′ is 2.3, the pK″ is 6.6, and the isoelectric point is 4.1. The addition of N-(carboxymethyl)chitosan to solutions (0.2–0.5mm) of transition-metal ions produces immediate insolubilization of N-(carboxymethyl)chitosan-metal ion chelates.     

Slow folding of a three-helix protein via a compact intermediate

The KIX domain of CREB binding protein (CBP) forms a small three-helix bundle which folds autonomously. Previous equilibrium unfolding experiments led to the suggestion that folding may not be strictly two-state. To investigate the folding mechanism in more detail, the folding kinetics of KIX have been studied by urea jump fluorescence-detected stopped-flow experiments. Clear evidence for an intermediate is obtained from the plot of the natural log of the observed rate constant versus denaturant concentration, the chevron plot, and from analysis of the initial fluorescence amplitudes of the stopped-flow experiments. The chevron plot exhibits a change in shape, rollover, at low denaturant concentrations, characteristic of the formation of an intermediate. The kinetic data can be fit to a three-state model involving a compact intermediate. An on-pathway model predicts that the position of the intermediate lies close to the native state. The folding rate in the absence of denaturant is 260 s(-)(1) at pH 7.5 and 25 degrees C. This is significantly slower than the rates of other helical proteins similar in size. The slow folding may be due to the necessity of forming a buried polar interaction in the native state. The potential functional significance of the folding intermediate is discussed.     

Mode of action of a family 75 chitosanase from Streptomyces avermitilis

Chitooligosaccharides (CHOS) are oligomers composed of glucosamine and N-acetylglucosamine with several interesting bioactivities that can be produced from enzymatic cleavage of chitosans. By controlling the degree of acetylation of the substrate chitosan, the enzyme, and the extent of enzyme degradation, CHOS preparations with limited variation in length and sequence can be produced. We here report on the degradation of chitosans with a novel family 75 chitosanase, SaCsn75A from Streptomyces avermitilis . By characterizing the CHOS preparations, we have obtained insight into the mode of action and subsite specificities of the enzyme. The degradation of a fully deacetylated and a 31% acetylated chitosan revealed that the enzyme degrade these substrates according to a nonprocessive, endo mode of action. With the 31% acetylated chitosan as substrate, the kinetics of the degradation showed an initial rapid phase, followed by a second slower phase. In the initial faster phase, an acetylated unit (A) is productively bound in subsite -1, whereas deacetylated units (D) are bound in the -2 subsite and the +1 subsite. In the slower second phase, D-units bind productively in the -1 subsite, probably with both acetylated and deacetylated units in the -2 subsite, but still with an absolute preference for deacetylated units in the +1 subsite. CHOS produced in the initial phase are composed of deacetylated units with an acetylated reducing end. In the slower second phase, higher amounts of low DP fully deacetylated oligomers (dimer and trimer) are produced, while the higher DP oligomers are dominated by compounds with acetylated reducing ends containing increasing amounts of internal acetylated units. The degradation of chitosans with varying degrees of acetylation to maximum extents of degradation showed that increasingly longer oligomers are produced with increasing degree of acetylation, and that the longer oligomers contain sequences of consecutive acetylated units interspaced by single deacetylated units. The catalytic properties of SaCsn75A differ from the properties of a previously characterized family 46 chitosanase from S. coelicolor (ScCsn46A).     

Activation enthalpy of diffusion for well fractionated dextrans in aqueous solutions

Interdiffusion coefficients have been determined for seven well defined dextran fractions in aqueous solutions at 20, 25, 30 and 35 degrees C. For dextrans with number average molecular weights (Mn) greater than about 20 000, a plot of the apparent activation energy of diffusion (ED) against Mn is given by: ED = 2.763 x 10(-5) Mn + 3.38. Similarly, for weigh molecular weight (Mw): ED = 1.889 x 10(-5) Mw + 3.61. The extrapolated ED values (3.38 and 3.61) are in reasonably good agreement with published data for the activation energy of self-diffusion for water.     

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.     

Rural to Urban Population Density Scaling of Crime and Property Transactions in English and Welsh Parliamentary Constituencies

Urban population scaling of resource use, creativity metrics, and human behaviors has been widely studied. These studies have not looked in detail at the full range of human environments which represent a continuum from the most rural to heavily urban. We examined monthly police crime reports and property transaction values across all 573 Parliamentary Constituencies in England and Wales, finding that scaling models based on population density provided a far superior framework to traditional population scaling. We found four types of scaling: i) non-urban scaling in which a single power law explained the relationship between the metrics and population density from the most rural to heavily urban environments, ii) accelerated scaling in which high population density was associated with an increase in the power-law exponent, iii) inhibited scaling where the urban environment resulted in a reduction in the power-law exponent but remained positive, and iv) collapsed scaling where transition to the high density environment resulted in a negative scaling exponent. Urban scaling transitions, when observed, took place universally between 10 and 70 people per hectare. This study significantly refines our understanding of urban scaling, making clear that some of what has been previously ascribed to urban environments may simply be the high density portion of non-urban scaling. It also makes clear that some metrics undergo specific transitions in urban environments and these transitions can include negative scaling exponents indicative of collapse. This study gives promise of far more sophisticated scale adjusted metrics and indicates that studies of urban scaling represent a high density subsection of overall scaling relationships which continue into rural environments.