Nonergodisity of a time series obeying L\'evy statistics

Time-averaged autocorrelation functions of a dichotomous random process switching between 1 and 0 and governed by wide power law sojourn time distribution are studied. Such a process, called a L\'evy walk, describes dynamical behaviors of many physical systems, fluorescence intermittency of semiconductor nanocrystals under continuous laser illumination being one example. When the mean sojourn time diverges the process is non-ergodic. In that case, the time average autocorrelation function is not equal to the ensemble averaged autocorrelation function, instead it remains random even in the limit of long measurement time. Several approximations for the distribution of this random autocorrelation function are obtained for different parameter ranges, and favorably compared to Monte Carlo simulations. Nonergodicity of the power spectrum of the process is briefly discussed, and a nonstationary Wiener-Khintchine theorem, relating the correlation functions and the power spectrum is presented. The considered situation is in full contrast to the usual assumptions of ergodicity and stationarity.Comment: 15 pages, 10 figure

Analysis of a microscopic stochastic model of microtubule dynamic instability

A novel theoretical model of dynamic instability of a system of linear (1D) microtubules (MTs) in a bounded domain is introduced for studying the role of a cell edge in vivo and analyzing the effect of competition for a limited amount of tubulin. The model differs from earlier models in that the evolution of MTs is based on the rates of single unit (e.g., a heterodimer per protofilament) transformations, in contrast to postulating effective rates/frequencies of larger-scale changes, extracted, e.g., from the length history plots of MTs. Spontaneous GTP hydrolysis with finite rate after polymerization is assumed, and theoretical estimates of an effective catastrophe frequency as well as other parameters characterizing MT length distributions and cap size are derived. We implement a simple cap model which does not include vectorial hydrolysis. We demonstrate that our theoretical predictions, such as steady state concentration of free tubulin, and parameters of MT length distributions, are in agreement with the numerical simulations. The present model establishes a quantitative link between microscopic parameters governing the dynamics of MTs and macroscopic characteristics of MTs in a closed system. Lastly, we use a computational Monte Carlo model to provide an explanation for non-exponential MT length distributions observed in experiments. In particular, we show that appearance of such non-exponential distributions in the experiments can occur because the true steady state has not been reached, and/or due to the presence of a cell edge.Comment: 14 pages, 7 figure

Brauer Groups of Local Elliptic and Hyperelliptic Curves and Central Division Algebras Over Their Function Fields

. We compute the torsion part of the unramified Brauer group of the function field of any elliptic curve defined over a local field of characteristc zero. For the case of a non dyadic ground field we prove that all elements of the 2-torsion part can be represented by quaternion algebras and describe them in an explicit form. We obtain a similar result for hyperelliptic curves with good reduction such that their defining equation y 2 = f(x) satisfies the condition deg f(x) 6j 0(mod 4). Contents 1 Introduction 2 2 Notations and preliminary results 3 3 Torsion of Brauer groups of curves and their Jacobians 13 4 The group 2 P ic 0 ( ¯ C)(k) 16 5 Presentation of elements of 2 Br C by central simple algebras (case of elliptic and hyperelliptic curves with good reduction) 19 6 Minimal Weierstrass equations of elliptic curves over non-dyadic fields 21 7 Torsion of local elliptic curves 27 8 The group 2 BrE. The case of good reduction 34 9 The group 2 Br E. The split case of bad reduction ..

Significant associations between driver gene mutations and DNA methylation alterations across many cancer types

<div><p>Recent evidence shows that mutations in several driver genes can cause aberrant methylation patterns, a hallmark of cancer. In light of these findings, we hypothesized that the landscapes of tumor genomes and epigenomes are tightly interconnected. We measured this relationship using principal component analyses and methylation-mutation associations applied at the nucleotide level and with respect to genome-wide trends. We found that a few mutated driver genes were associated with genome-wide patterns of aberrant hypomethylation or CpG island hypermethylation in specific cancer types. In addition, we identified associations between 737 mutated driver genes and site-specific methylation changes. Moreover, using these mutation-methylation associations, we were able to distinguish between two uterine and two thyroid cancer subtypes. The driver gene mutation–associated methylation differences between the thyroid cancer subtypes were linked to differential gene expression in JAK-STAT signaling, NADPH oxidation, and other cancer-related pathways. These results establish that driver gene mutations are associated with methylation alterations capable of shaping regulatory network functions. In addition, the methodology presented here can be used to subdivide tumors into more homogeneous subsets corresponding to underlying molecular characteristics, which could improve treatment efficacy.</p></div

Associations between mutated driver genes and HyperZ and HypoZ indices or site–specific methylation alterations.

<p>Associations between mutated driver genes and HyperZ and HypoZ indices or site–specific methylation alterations.</p

Proportion of positive and negative associations with methylation for 17 recurrently mutated driver genes.

<p>(A) Bar plots show the proportion of methylation probes for each driver gene (<i>labels at bottom</i>) and cancer type (<i>labels at top</i>) displaying positive and negative associations. Positive associations are plotted above the horizontal line, negative associations below the horizontal line. Associations are further stratified by CpG subset: CpG islands (CGI), shores and shelves (SS), and open sea (regions outside CGIs and SSs). Driver genes were classified into three groups based on the directionality of their predominant associations (<i>negative</i>, <i>positive</i>, <i>inconsistent</i>). All genes shown were associated with more than 1,000 probes, in at least two cancer types. See <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005840#pcbi.1005840.t001" target="_blank">Table 1</a> for cancer type abbreviations. (B) Plotted as in (A), using: (1) positively associated and hypermethylated probes and (2) negatively associated and hypomethylated probes. *Hyper- or hypomethylated probes were not identified for glioblastoma (GBM), stomach adenocarcinoma (STAD), skin cutaneous melanoma (SKCM), and testicular germ cell tumor (TGCT) due to a lack of normal samples.</p

Driver gene–methylation associations and CpG subsets.

<p>(A) The total number of probes associated with any driver gene is shown for each cancer type (<i>top of each column</i>). Each point represents the fraction of corresponding probes associated with a driver gene (y-axis). Names are shown for each of the top three driver genes if they account for more than 10% of total probes (<i>dotted line</i>). See <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005840#pcbi.1005840.t001" target="_blank">Table 1</a> for cancer type abbreviations. (B) Driver genes with the most probe associations in each cancer type (<i>gene names in panel</i>). The bar plots show the proportion of associated probes in each of the three CpG subsets [CpG islands (CGIs), shores and shelves (SSs), or open sea], stratified by the direction of association (+/-). Dashed lines indicate the divisions expected if associations were proportionally distributed. No probes were associated with driver genes in BLCA, LUSC, and READ.</p