Quantum Analog of Shannon's Lower Bound Theorem

Shannon proved that almost all Boolean functions require a circuit of size $\Theta(2^n/n)$. We prove a quantum analog of this classical result. Unlike in the classical case the number of quantum circuits of any fixed size that we allow is uncountably infinite. Our main tool is a classical result in real algebraic geometry bounding the number of realizable sign conditions of any finite set of real polynomials in many variables.Comment: Comments welcom

Detection of subtle variations as consensus motifs

AbstractWe address the problem of detecting consensus motifs, that occur with subtle variations, across multiple sequences. These are usually functional domains in DNA sequences such as transcriptional binding factors or other regulatory sites. The problem in its generality has been considered difficult and various benchmark data serve as the litmus test for different computational methods. We present a method centered around unsupervised combinatorial pattern discovery. The parameters are chosen using a careful statistical analysis of consensus motifs. This method works well on the benchmark data and is general enough to be extended to a scenario where the variation in the consensus motif includes indels (along with mutations). We also present some results on detection of transcription binding factors in human DNA sequences

10231 Abstracts Collection -- Structure Discovery in Biology: Motifs, Networks & Phylogenies

From 06.06. to 11.06.2010, the Dagstuhl Seminar 10231 ``Structure Discovery in Biology: Motifs, Networks & Phylogenies \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Essential Simplices in Persistent Homology and Subtle Admixture Detection

We introduce a robust mathematical definition of the notion of essential elements in a basis of the homology space and prove that these elements are unique. Next we give a novel visualization of the essential elements of the basis of the homology space through a rainfall-like plot (RFL). This plot is data-centric, i.e., is associated with the individual samples of the data, as opposed to the structure-centric barcodes of persistent homology. The proof-of-concept was tested on data generated by SimRA that simulates different admixture scenarios. We show that the barcode analysis can be used not just to detect the presence of admixture but also estimate the number of admixed populations. We also demonstrate that data-centric RFL plots have the potential to further disentangle the common history into admixture events and relative timing of the events, even in very complex scenarios

Combinatorial Pattern Discovery Approach for the Folding Trajectory Analysis of a β-Hairpin

The study of protein folding mechanisms continues to be one of the most challenging problems in computational biology. Currently, the protein folding mechanism is often characterized by calculating the free energy landscape versus various reaction coordinates, such as the fraction of native contacts, the radius of gyration, RMSD from the native structure, and so on. In this paper, we present a combinatorial pattern discovery approach toward understanding the global state changes during the folding process. This is a first step toward an unsupervised (and perhaps eventually automated) approach toward identification of global states. The approach is based on computing biclusters (or patterned clusters)—each cluster is a combination of various reaction coordinates, and its signature pattern facilitates the computation of the Z-score for the cluster. For this discovery process, we present an algorithm of time complexity c∈RO((N + nm) log n), where N is the size of the output patterns and (n × m) is the size of the input with n time frames and m reaction coordinates. To date, this is the best time complexity for this problem. We next apply this to a β-hairpin folding trajectory and demonstrate that this approach extracts crucial information about protein folding intermediate states and mechanism. We make three observations about the approach: (1) The method recovers states previously obtained by visually analyzing free energy surfaces. (2) It also succeeds in extracting meaningful patterns and structures that had been overlooked in previous works, which provides a better understanding of the folding mechanism of the β-hairpin. These new patterns also interconnect various states in existing free energy surfaces versus different reaction coordinates. (3) The approach does not require calculating the free energy values, yet it offers an analysis comparable to, and sometimes better than, the methods that use free energy landscapes, thus validating the choice of reaction coordinates. (An abstract version of this work was presented at the 2005 Asia Pacific Bioinformatics Conference [1].

Sampling ARG of multiple populations under complex configurations of subdivision and admixture.

Abstract Motivation: Simulating complex evolution scenarios of multiple populations is an important task for answering many basic questions relating to population genomics. Apart from the population samples, the underlying Ancestral Recombinations Graph (ARG) is an additional important means in hypothesis checking and reconstruction studies. Furthermore, complex simulations require a plethora of interdependent parameters making even the scenario-specification highly non-trivial. Results: We present an algorithm SimRA that simulates generic multiple population evolution model with admixture. It is based on random graphs that improve dramatically in time and space requirements of the classical algorithm of single populations. Using the underlying random graphs model, we also derive closed forms of expected values of the ARG characteristics i.e., height of the graph, number of recombinations, number of mutations and population diversity in terms of its defining parameters. This is crucial in aiding the user to specify meaningful parameters for the complex scenario simulations, not through trial-and-error based on raw compute power but intelligent parameter estimation. To the best of our knowledge this is the first time closed form expressions have been computed for the ARG properties. We show that the expected values closely match the empirical values through simulations. Finally, we demonstrate that SimRA produces the ARG in compact forms without compromising any accuracy. We demonstrate the compactness and accuracy through extensive experiments. Availability and implementation: SimRA (Simulation based on Random graph Algorithms) source, executable, user manual and sample input-output sets are available for downloading at: https://github.com/ComputationalGenomics/SimRA Contact: [email protected] Supplementary information: Supplementary data are available at Bioinformatics online