The entropy distance between the Wiener and stationary Gaussian measures

Investigating the entropy distance between the Wiener measure, W-t0,W- (tau) and stationary Gaussian measures, Q(t0, tau) on the space of continuous functions C[t(0) - tau, t(0) + tau], we show that in some cases this distance can essentially be computed. This is done by explicitly computing a related quantity which in effect is a valid approximation of the entropy distance, provided it is sufficiently small; this will be the case if tau/t(0) is small. We prove that H(Wt(0, tau), Q(t0, tau)) > tau/2(t0), and then show that tau/2t(0) is essentially the typical case of such entropy distance, provided the mean and the variance of the stationary measures are set "appropriately". Utilizing a similar technique, we estimate the entropy distance between the Ornstein-Uhlenbeck measure and other stationary Gaussian measures on C[1 - tau, 1 + tau]. Using this result combined with a variant of the triangle inequality for the entropy distance, which we devise, yields an upper bound on the entropy distance between stationary Gaussian measures which are absolutely continuous with respect to the Wiener measure

Absolute continuity between the Wiener and stationary Gaussian measures

It is known that the entropy distance between two Gaussian measures is finite if, and only if, they are absolutely continuous with respect to one another. Shepp (1966) characterized the correlations corresponding to stationary Gaussian measures that are absolutely continuous with respect to the Wiener measure. By analyzing the entropy distance, we show that one of his conditions, involving the spectrum of an associated operator, is essentially extraneous, providing a simple criterion for finite entropy distance in this case

On L^p Bounds for Kakeya Maximal Functions and the Minkowski Dimension in R^2

We prove that the bound on the L^p norms of the Kakeya type maximal functions studied by Cordoba [2] and Bourgain [1] are sharp for p > 2. The proof is based on a construction originally due to Schoenberg [5], for which we provide an alternative derivation. We also show that r^2 log (1/r) is the exact Minkowski dimension of the class of Kakeya sets in R^2, and prove that the exact Hausdorff dimension of these sets is between r^2 log (1/r) and r^2 log (1/r) [log log (1/r)]^(2+ε)

A particle swarm optimization-based algorithm for finding gapped motifs

<p>Abstract</p> <p>Background</p> <p>Identifying approximately repeated patterns, or motifs, in DNA sequences from a set of co-regulated genes is an important step towards deciphering the complex gene regulatory networks and understanding gene functions.</p> <p>Results</p> <p>In this work, we develop a novel motif finding algorithm (PSO+) using a population-based stochastic optimization technique called Particle Swarm Optimization (PSO), which has been shown to be effective in optimizing difficult multidimensional problems in continuous domains. We propose a modification of the standard PSO algorithm to handle discrete values, such as characters in DNA sequences. The algorithm provides several features. First, we use both consensus and position-specific weight matrix representations in our algorithm, taking advantage of the efficiency of the former and the accuracy of the latter. Furthermore, many real motifs contain gaps, but the existing methods usually ignore them or assume a user know their exact locations and lengths, which is usually impractical for real applications. In comparison, our method models gaps explicitly, and provides an easy solution to find gapped motifs without any detailed knowledge of gaps. Our method allows the presence of input sequences containing zero or multiple binding sites.</p> <p>Conclusion</p> <p>Experimental results on synthetic challenge problems as well as real biological sequences show that our method is both more efficient and more accurate than several existing algorithms, especially when gaps are present in the motifs.</p

Novel features of ARS selection in budding yeast Lachancea kluyveri

<p>Abstract</p> <p>Background</p> <p>The characterization of DNA replication origins in yeast has shed much light on the mechanisms of initiation of DNA replication. However, very little is known about the evolution of origins or the evolution of mechanisms through which origins are recognized by the initiation machinery. This lack of understanding is largely due to the vast evolutionary distances between model organisms in which origins have been examined.</p> <p>Results</p> <p>In this study we have isolated and characterized autonomously replicating sequences (ARSs) in <it>Lachancea kluyveri </it>- a pre-whole genome duplication (WGD) budding yeast. Through a combination of experimental work and rigorous computational analysis, we show that <it>L. kluyveri </it>ARSs require a sequence that is similar but much longer than the ARS Consensus Sequence well defined in <it>Saccharomyces cerevisiae</it>. Moreover, compared with <it>S. cerevisiae </it>and <it>K. lactis</it>, the replication licensing machinery in <it>L. kluyveri </it>seems more tolerant to variations in the ARS sequence composition. It is able to initiate replication from almost all <it>S. cerevisiae </it>ARSs tested and most <it>Kluyveromyces lactis </it>ARSs. In contrast, only about half of the <it>L. kluyveri </it>ARSs function in <it>S. cerevisiae </it>and less than 10% function in <it>K. lactis</it>.</p> <p>Conclusions</p> <p>Our findings demonstrate a replication initiation system with novel features and underscore the functional diversity within the budding yeasts. Furthermore, we have developed new approaches for analyzing biologically functional DNA sequences with ill-defined motifs.</p

A Comprehensive Genome-Wide Map of Autonomously Replicating Sequences in a Naive Genome

Eukaryotic chromosomes initiate DNA synthesis from multiple replication origins. The machinery that initiates DNA synthesis is highly conserved, but the sites where the replication initiation proteins bind have diverged significantly. Functional comparative genomics is an obvious approach to study the evolution of replication origins. However, to date, the Saccharomyces cerevisiae replication origin map is the only genome map available. Using an iterative approach that combines computational prediction and functional validation, we have generated a high-resolution genome-wide map of DNA replication origins in Kluyveromyces lactis. Unlike other yeasts or metazoans, K. lactis autonomously replicating sequences (KlARSs) contain a 50 bp consensus motif suggestive of a dimeric structure. This motif is necessary and largely sufficient for initiation and was used to dependably identify 145 of the up to 156 non-repetitive intergenic ARSs projected for the K. lactis genome. Though similar in genome sizes, K. lactis has half as many ARSs as its distant relative S. cerevisiae. Comparative genomic analysis shows that ARSs in K. lactis and S. cerevisiae preferentially localize to non-syntenic intergenic regions, linking ARSs with loci of accelerated evolutionary change

Functional Characterization of Transcription Factor Motifs Using Cross-species Comparison across Large Evolutionary Distances

We address the problem of finding statistically significant associations between cis-regulatory motifs and functional gene sets, in order to understand the biological roles of transcription factors. We develop a computational framework for this task, whose features include a new statistical score for motif scanning, the use of different scores for predicting targets of different motifs, and new ways to deal with redundancies among significant motif–function associations. This framework is applied to the recently sequenced genome of the jewel wasp, Nasonia vitripennis, making use of the existing knowledge of motifs and gene annotations in another insect genome, that of the fruitfly. The framework uses cross-species comparison to improve the specificity of its predictions, and does so without relying upon non-coding sequence alignment. It is therefore well suited for comparative genomics across large evolutionary divergences, where existing alignment-based methods are not applicable. We also apply the framework to find motifs associated with socially regulated gene sets in the honeybee, Apis mellifera, using comparisons with Nasonia, a solitary species, to identify honeybee-specific associations