The basic reproductive ratio of a word, the maximum size of a lexicon

Language is about words and rules. While there is some discussion to what extent rules are learned or innate, it is clear that words have to be learned. Here I construct a mathematical framework for the population dynamics of language evolution with particular emphasis on how words are propagated over generations. I define the basic reproductive ratio of word, R, and show that R > 1 is required for words to be maintained in the lexicon of a language. Assuming that the frequency distribution of words follow Zipf's law, an upper limit is obtained for the number of words in a language that relies exclusively on oral transmission.     

Zipf's law in importance of genes for cancer classification using microarray data

Using a measure of how differentially expressed a gene is in two biochemically/phenotypically different conditions, we can rank all genes in a microarray dataset. We have shown that the falling-off of this measure (normalized maximum likelihood in a classification model such as logistic regression) as a function of the rank is typically a power-law function. This power-law function in other similar ranked plots are known as the Zipf's law, observed in many natural and social phenomena. The presence of this power-law function prevents an intrinsic cutoff point between the "important" genes and "irrelevant" genes. We have shown that similar power-law functions are also present in permuted dataset, and provide an explanation from the well-known chi(2) distribution of likelihood ratios. We discuss the implication of this Zipf's law on gene selection in a microarray data analysis, as well as other characterizations of the ranked likelihood plots such as the rate of fall-off of the likelihood.     

When language breaks into pieces. A conflict between communication through isolated signals and language

Here, we study a communication model where signals associate to stimuli. The model assumes that signals follow Zipf's law and the exponent of the law depends on a balance between maximizing the information transfer and saving the cost of signal use. We study the effect of tuning that balance on the structure of signal-stimulus associations. The model starts from two recent results. First, the exponent grows as the weight of information transfer increases. Second, a rudimentary form of language is obtained when the network of signal-stimulus associations is almost connected. Here, we show the existence of a sudden destruction of language once a critical balance is crossed. The model shows that maximizing the information transfer through isolated signals and language are in conflict. The model proposes a strong reason for not finding large exponents in complex communication systems: language is in danger. Besides, the findings suggest that human words may need to be ambiguous to keep language alive. Interestingly, the model predicts that large exponents should be associated to decreased synaptic density. It is not surprising that the largest exponents correspond to schizophrenic patients since, according to the spirit of Feinberg's hypothesis, i.e. decreased synaptic density may lead to schizophrenia. Our findings suggest that the exponent of Zipf's law is intimately related to language and that it could be used to detect anomalous structure and organization of the brain.     

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.     

The language of gene ontology: a Zipf's law analysis

ABSTRACT: BACKGROUND: Most major genome projects and sequence databases provide a GO annotation of their data,either automatically or through human annotators, creating a large corpus of data written inthe language of GO. Texts written in natural language show a statistical power law behaviour,Zipf's law, the exponent of which can provide useful information on the nature of thelanguage being used. We have therefore explored the hypothesis that collections of GOannotations will show similar statistical behaviours to natural language. RESULTS: Annotations from the Gene Ontology Annotation project were found to follow Zipf's law.Surprisingly, the measured power law exponents were consistently different betweenannotation captured using the three GO sub-ontologies in the corpora (function, process andcomponent). On filtering the corpora using GO evidence codes we found that the value of themeasured power law exponent responded in a predictable way as a function of the evidencecodes used to support the annotation. CONCLUSIONS: Techniques from computational linguistics can provide new insights into the annotationprocess. GO annotations show similar statistical behaviours to those seen in natural languagewith measured exponents that provide a signal which correlates with the nature of the evidence codes used to support the annotations, suggesting that the measured exponent mightprovide a signal regarding the information content of the annotation.     

A Cart for Cardiopulmonary Resuscitation

The authors have devised a cart for use during cardiopulmonary resuscitation, the prime purpose of which is to bring every essential piece of equipment, and all the drugs generally required, to the immediate area to revive a pulseless and/or apneic patient with the least effort and confusion. The strategically placed contents of the cart are described.The regular use of such a cart, following appropriate formal staff training and instruction in resuscitation procedures, and frequent reading of an appropriate instruction poster, as well as the reporting of all these events on an appropriate protocol, will increase the effectiveness of acute resuscitation in hospital.     

Can Zipf's law be adapted to normalize microarrays?

Background  

Normalization is the process of removing non-biological sources of variation between array experiments. Recent investigations of data in gene expression databases for varying organisms and tissues have shown that the majority of expressed genes exhibit a power-law distribution with an exponent close to -1 (i.e. obey Zipf's law). Based on the observation that our single channel and two channel microarray data sets also followed a power-law distribution, we were motivated to develop a normalization method based on this law, and examine how it compares with existing published techniques. A computationally simple and intuitively appealing technique based on this observation is presented.     

An empirical test of Lanchester's square law: mortality during battles of the fire ant Solenopsis invicta

Lanchester's models of attrition describe casualty rates during battles between groups as functions of the numbers of individuals and their fighting abilities. Originally developed to describe human warfare, Lanchester's square law has been hypothesized to apply broadly to social animals as well, with important consequences for their aggressive behaviour and social structure. According to the square law, the fighting ability of a group is proportional to the square of the number of individuals, but rises only linearly with fighting ability of individuals within the group. By analyzing mortality rates of fire ants (Solenopsis invicta) fighting in different numerical ratios, we provide the first quantitative test of Lanchester's model for a non-human animal. Casualty rates of fire ants were not consistent with the square law; instead, group fighting ability was an approximately linear function of group size. This implies that the relative numbers of casualties incurred by two fighting groups are not strongly affected by relative group sizes and that battles do not disproportionately favour group size over individual prowess.     

A Note on the Use of Unbiased Estimating Equations to Estimate Correlation in Analysis of Longitudinal Trials

Longitudinal trials can yield outcomes that are continuous, binary (yes/no), or are realizations of counts. In this setting we compare three approaches that have been proposed for estimation of the correlation in the framework of generalized estimating equations (GEE): quasi‐least squares (QLS), pseudo‐likelihood (PL), and an approach we refer to as Wang–Carey (WC). We prove that WC and QLS are identical for the first‐order autoregressive AR(1) correlation structure. Using simulations, we then develop guidelines for selection of an appropriate method for analysis of data from a longitudinal trial. In particular, we argue that no method is uniformly superior for analysis of unbalanced and unequally spaced data with a Markov correlation structure. Choice of the best approach will depend on the degree of imbalance and variability in the temporal spacing of measurements, value of the correlation, and type of outcome, e.g. binary or continuous. Finally, we contrast the methods in analysis of a longitudinal study of obesity following renal transplantation in children (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Grassland harvesting alters ant community trophic structure: An isotopic study in tallgrass prairies

Disturbances have long been recognized as important forces for structuring natural communities but their effects on trophic structure are not well understood, particularly in terrestrial systems. This is in part because quantifying trophic linkages is a challenge, especially for small organisms with cryptic feeding behaviors such as insects, and often relies on conducting labor‐intensive feeding trials or extensive observations in the field. In this study, we used stable isotopes of carbon and nitrogen to examine how disturbance (annual biomass harvesting) in tallgrass prairies affected the trophic position, trophic range, and niche space of ants, a widespread grassland consumer. We hypothesized that biomass harvest would remove important food and nesting resources of insects thus affecting ant feeding relationships and trophic structure. We found shifts in the feeding relationships inferred by isotopic signatures with harvest. In particular, these shifts suggest that ants within harvest sites utilized resources at lower trophic levels (possibly plant‐based resources or herbivores), expanded trophic breadth, and occupied different niche spaces. Shifts in resource use following harvest could be due to harvest‐mediated changes in both the plant and arthropod communities that might affect the strength of competition or alter plant nitrogen availability. Because shifts in resource use alter the flow of nutrients across the food web, disturbance effects on ants could have ecosystem‐level consequences through nutrient cycling.     

Identification of human proteins using the linguist's tools

The symbolic sequences of the exons that make human proteins are subjected to methods of statistical linguistics. The ideas developed for the natural languages by G. K. Zipf, when applied to these sequences, show significant promise. In particular, we argue, the Zipf's exponent differentiates, and hence, identifies disparate human sequences.     

同义密码子使用模式对蛋白产物表达及构象形成的影响*

鉴于遗传密码子的简并性能够将基因遗传信息的容量提升,同义密码子使用偏嗜性得以在生物体的基因组中广泛存在。虽然同义密码子之间碱基的变化并不能导致氨基酸种类的改变,在研究mRNA半衰期、编码多肽翻译效率及肽链空间构象正确折叠的准确性和翻译等这一系列过程中发现,同义密码子使用的偏嗜性在某种程度上通过精微调控翻译机制体现其遗传学功能。同义密码子指导tRNA在翻译过程中识别核糖体的速率变化是由氨基酸的特定顺序决定,并且在新生多肽链合成时,蛋白质共翻译转运机制同时调节其空间构象的正确折叠从而保证蛋白的正常生物学功能。某些同义密码子使用偏嗜性与特定蛋白结构的形成具有显著相关性,密码子使用偏嗜性一旦改变将可能导致新生多肽空间构象出现错误折叠。结合近些年来国内外在此领域的研究成果,阐述同义密码子使用偏嗜性如何发挥精微调控翻译的生物学功能与作用。     

Specific, non-nutritional association between an ascomycete fungus and Allomerus plant-ants

Ant-fungus associations are well known from attine ants, whose nutrition is based on a symbiosis with basidiomycete fungi. Otherwise, only a few non-nutritional ant-fungus associations have been recorded to date. Here we focus on one of these associations involving Allomerus plant-ants that build galleried structures on their myrmecophytic hosts in order to ambush prey. We show that this association is not opportunistic because the ants select from a monophyletic group of closely related fungal haplotypes of an ascomycete species from the order Chaetothyriales that consistently grows on and has been isolated from the galleries. Both the ants' behaviour and an analysis of the genetic population structure of the ants and the fungus argue for host specificity in this interaction. The ants' behaviour reveals a major investment in manipulating, growing and cleaning the fungus. A molecular analysis of the fungus demonstrates the widespread occurrence of one haplotype and many other haplotypes with a lower occurrence, as well as significant variation in the presence of these fungal haplotypes between areas and ant species. Altogether, these results suggest that such an interaction might represent an as-yet undescribed type of specific association between ants and fungus in which the ants cultivate fungal mycelia to strengthen their hunting galleries.     

Contribution to knowledge of the morphogenesis of the nasal apparatus of the fallow deer (Dama dama L.)

As in the red deer, in the fallow deer embryo we found a number of ancestral structures reminiscent of relationships in other mammals, such as paraseptal cartilages, a septum nasi with trabecular widening, a lamina transversalis ant., a cart. ectochoanalis, a capsule wall with a roof and a lateral wall formed of a clearly distinguishable cart. parietotectalis and cart. paranasalis, an ethmoturbinale I projecting a long way rostrally and additionally, in the fallow deer, cart. paraseptales posteriores. I regard the relationship of the cart. alaris inf. to the parietotectal cartilage (or "marginoturbinale") as relatively "primitive"; this may mean that the term "atrioturbinale" is also justified in mammals and that the relevant structure is homologous with the one known by the same name in birds. The specializations found during study of the morphogenesis of the nasal apparatus in the red deer (Slaby 1990b) are accentuated in the fallow deer. The chief ones are the specific rostral processes of the anlage of the nasal septum, which are a significant part of reinforcement of the nostril, the marked widening of the nasal capsule in a lateral direction (so that even the paranasal cartilages have a largely horizontal course), the striking ventrolateral bulge in the nasal capsule at the beginning of the olfactory region and the final resultant decrease in the height (i. e. flattening) of the capsule. This leads to reduction of the frontoturbinalia and their corresponding recesses, which - where they are developed - are oriented more horizontally. The structure of ethmoturbinale I, together with its insertion, is also simplified. As in the corresponding red deer embryo, the paranasal cartilage zone in the anterior part of the olfactory region is strikingly thickened; the frontoturbinalia do not, however, originate (in our stage) by the formation of cavities in the cartilage, but develop as simple processes. A crista semicircularis and foramen epiphaniale and also, as distinct from the red deer embryo, cart. paraseptales posteriores, are clearly discernible. In conclusion, it can therefore be claimed that the morphogenesis of specialized cervid features is accentuated in Dama more than in Cervus and that relationships in the fallow deer represent a further step in specialization, or - if we are speaking of the development of radiations - specialization here has progressed further.     

Structural diversity of protein segments follows a power-law distribution

The local structures of protein segments were classified and their distribution was analyzed to explore the structural diversity of proteins. Representative proteins were divided into short segments using a sliding L-residue window. Each set of local structures consisting of consecutive 1-31 amino acids was classified using a single-pass clustering method. The results demonstrate that the local structures of proteins are very unevenly distributed in the protein universe. The distribution of local structures of relatively long segments shows a power-law behavior that is formulated well by Zipf's law, implying that a protein structure possesses recursive and fractal characteristics. The degree of effective conformational freedom per residue as well as the structure entropy per residue decreases gradually with an increasing value of L and then converges to constant values. This suggests that the number of protein conformations resides within the range between 1.2L and 1.5L and that 10- to 20-residue segments are already proteinlike in terms of their structural diversity.     

Zipf's law organizes a psychiatric ward.

We developed a simple mathematical model based on power law fitting for describing the interactions among patients from a psychiatric ward. First we defined a protocol in order to evaluate in a quantative way the state of the patient, measuring sociability/restlessness through a daily analysis of the behavior and attributing a grade for both parameters, per patient. The grades were checked by two different specialists and a table of incidence was constructed. This table generated power laws for the grades and their variations. We concluded that power laws, like Zipf's law, may be good to explain the data, showing a self-organizing process that indicates a strong interaction component determining the whole behavior. We would like to see more data being collected, in other centers and among normal populations, trying to quantify complex collective behavioral phenomena using self-organizing criticality laws.