Moments of the Count of a Regular Expression in a Heterogeneous Random Sequence

International audienceWe focus here on the distribution of the random count N of a regular expression in a multi-state random sequence generated by a heterogenous Markov source. We first briefly recall how classical Markov chain embedding techniques allow reducing the problem to the count of specific transitions in a (heterogenous) order 1 Markov chain over a deterministic finite automaton state space. From this result we derive the expression of both the mgf/pgf of N as well as the factorial and non-factorial moments of N. We then introduce the notion of evidence-based constraints in this context. Following the classical forward/backward algorithm in hidden Markov models, we provide explicit recursions allowing to compute the mgf/pgf of N under the evidence constraint. All the results presented are illustrated with a toy example

Effective p-value computations using Finite Markov Chain Imbedding (FMCI): application to local score and to pattern statistics

The technique of Finite Markov Chain Imbedding (FMCI) is a classical approach to complex combinatorial problems related to sequences. In order to get efficient algorithms, it is known that such approaches need to be first rewritten using recursive relations. We propose here to give here a general recursive algorithms allowing to compute in a numerically stable manner exact Cumulative Distribution Function (CDF) or complementary CDF (CCDF). These algorithms are then applied in two particular cases: the local score of one sequence and pattern statistics. In both cases, asymptotic developments are derived. For the local score, our new approach allows for the very first time to compute exact p-values for a practical study (finding hydrophobic segments in a protein database) where only approximations were available before. In this study, the asymptotic approximations appear to be completely unreliable for 99.5% of the considered sequences. Concerning the pattern statistics, the new FMCI algorithms dramatically outperform the previous ones as they are more reliable, easier to implement, faster and with lower memory requirements

Exact distribution of a pattern in a set of random sequences generated by a Markov source: applications to biological data

<p>Abstract</p> <p>Background</p> <p>In bioinformatics it is common to search for a pattern of interest in a potentially large set of rather short sequences (upstream gene regions, proteins, exons, etc.). Although many methodological approaches allow practitioners to compute the distribution of a pattern count in a random sequence generated by a Markov source, no specific developments have taken into account the counting of occurrences in a set of independent sequences. We aim to address this problem by deriving efficient approaches and algorithms to perform these computations both for low and high complexity patterns in the framework of homogeneous or heterogeneous Markov models.</p> <p>Results</p> <p>The latest advances in the field allowed us to use a technique of optimal Markov chain embedding based on deterministic finite automata to introduce three innovative algorithms. Algorithm 1 is the only one able to deal with heterogeneous models. It also permits to avoid any product of convolution of the pattern distribution in individual sequences. When working with homogeneous models, Algorithm 2 yields a dramatic reduction in the complexity by taking advantage of previous computations to obtain moment generating functions efficiently. In the particular case of low or moderate complexity patterns, Algorithm 3 exploits power computation and binary decomposition to further reduce the time complexity to a logarithmic scale. All these algorithms and their relative interest in comparison with existing ones were then tested and discussed on a toy-example and three biological data sets: structural patterns in protein loop structures, PROSITE signatures in a bacterial proteome, and transcription factors in upstream gene regions. On these data sets, we also compared our exact approaches to the tempting approximation that consists in concatenating the sequences in the data set into a single sequence.</p> <p>Conclusions</p> <p>Our algorithms prove to be effective and able to handle real data sets with multiple sequences, as well as biological patterns of interest, even when the latter display a high complexity (PROSITE signatures for example). In addition, these exact algorithms allow us to avoid the edge effect observed under the single sequence approximation, which leads to erroneous results, especially when the marginal distribution of the model displays a slow convergence toward the stationary distribution. We end up with a discussion on our method and on its potential improvements.</p

Mining protein loops using a structural alphabet and statistical exceptionality

<p>Abstract</p> <p>Background</p> <p>Protein loops encompass 50% of protein residues in available three-dimensional structures. These regions are often involved in protein functions, e.g. binding site, catalytic pocket... However, the description of protein loops with conventional tools is an uneasy task. Regular secondary structures, helices and strands, have been widely studied whereas loops, because they are highly variable in terms of sequence and structure, are difficult to analyze. Due to data sparsity, long loops have rarely been systematically studied.</p> <p>Results</p> <p>We developed a simple and accurate method that allows the description and analysis of the structures of short and long loops using structural motifs without restriction on loop length. This method is based on the structural alphabet HMM-SA. HMM-SA allows the simplification of a three-dimensional protein structure into a one-dimensional string of states, where each state is a four-residue prototype fragment, called structural letter. The difficult task of the structural grouping of huge data sets is thus easily accomplished by handling structural letter strings as in conventional protein sequence analysis. We systematically extracted all seven-residue fragments in a bank of 93000 protein loops and grouped them according to the structural-letter sequence, named structural word. This approach permits a systematic analysis of loops of all sizes since we consider the structural motifs of seven residues rather than complete loops. We focused the analysis on highly recurrent words of loops (observed more than 30 times). Our study reveals that 73% of loop-lengths are covered by only 3310 highly recurrent structural words out of 28274 observed words). These structural words have low structural variability (mean RMSd of 0.85 Å). As expected, half of these motifs display a flanking-region preference but interestingly, two thirds are shared by short (less than 12 residues) and long loops. Moreover, half of recurrent motifs exhibit a significant level of amino-acid conservation with at least four significant positions and 87% of long loops contain at least one such word. We complement our analysis with the detection of statistically over-represented patterns of structural letters as in conventional DNA sequence analysis. About 30% (930) of structural words are over-represented, and cover about 40% of loop lengths. Interestingly, these words exhibit lower structural variability and higher sequential specificity, suggesting structural or functional constraints.</p> <p>Conclusions</p> <p>We developed a method to systematically decompose and study protein loops using recurrent structural motifs. This method is based on the structural alphabet HMM-SA and not on structural alignment and geometrical parameters. We extracted meaningful structural motifs that are found in both short and long loops. To our knowledge, it is the first time that pattern mining helps to increase the signal-to-noise ratio in protein loops. This finding helps to better describe protein loops and might permit to decrease the complexity of long-loop analysis. Detailed results are available at <url>http://www.mti.univ-paris-diderot.fr/publication/supplementary/2009/ACCLoop/</url>.</p

Future Contingents and the Logic of Temporal Omniscience

At least since Aristotle’s famous 'sea-battle' passages in On Interpretation 9, some substantial minority of philosophers has been attracted to the doctrine of the open future--the doctrine that future contingent statements are not true. But, prima facie, such views seem inconsistent with the following intuition: if something has happened, then (looking back) it was the case that it would happen. How can it be that, looking forwards, it isn’t true that there will be a sea battle, while also being true that, looking backwards, it was the case that there would be a sea battle? This tension forms, in large part, what might be called the problem of future contingents. A dominant trend in temporal logic and semantic theorizing about future contingents seeks to validate both intuitions. Theorists in this tradition--including some interpretations of Aristotle, but paradigmatically, Thomason (1970), as well as more recent developments in Belnap, et. al (2001) and MacFarlane (2003, 2014)--have argued that the apparent tension between the intuitions is in fact merely apparent. In short, such theorists seek to maintain both of the following two theses: (i) the open future: Future contingents are not true, and (ii) retro-closure: From the fact that something is true, it follows that it was the case that it would be true. It is well-known that reflection on the problem of future contingents has in many ways been inspired by importantly parallel issues regarding divine foreknowledge and indeterminism. In this paper, we take up this perspective, and ask what accepting both the open future and retro-closure predicts about omniscience. When we theorize about a perfect knower, we are theorizing about what an ideal agent ought to believe. Our contention is that there isn’t an acceptable view of ideally rational belief given the assumptions of the open future and retro-closure, and thus this casts doubt on the conjunction of those assumptions

Should We Abandon the t-Test in the Analysis of Gene Expression Microarray Data: A Comparison of Variance Modeling Strategies

High-throughput post-genomic studies are now routinely and promisingly investigated in biological and biomedical research. The main statistical approach to select genes differentially expressed between two groups is to apply a t-test, which is subject of criticism in the literature. Numerous alternatives have been developed based on different and innovative variance modeling strategies. However, a critical issue is that selecting a different test usually leads to a different gene list. In this context and given the current tendency to apply the t-test, identifying the most efficient approach in practice remains crucial. To provide elements to answer, we conduct a comparison of eight tests representative of variance modeling strategies in gene expression data: Welch's t-test, ANOVA [1], Wilcoxon's test, SAM [2], RVM [3], limma [4], VarMixt [5] and SMVar [6]. Our comparison process relies on four steps (gene list analysis, simulations, spike-in data and re-sampling) to formulate comprehensive and robust conclusions about test performance, in terms of statistical power, false-positive rate, execution time and ease of use. Our results raise concerns about the ability of some methods to control the expected number of false positives at a desirable level. Besides, two tests (limma and VarMixt) show significant improvement compared to the t-test, in particular to deal with small sample sizes. In addition limma presents several practical advantages, so we advocate its application to analyze gene expression data

An analysis of single amino acid repeats as use case for application specific background models

Background Sequence analysis aims to identify biologically relevant signals against a backdrop of functionally meaningless variation. Increasingly, it is recognized that the quality of the background model directly affects the performance of analyses. State-of-the-art approaches rely on classical sequence models that are adapted to the studied dataset. Although performing well in the analysis of globular protein domains, these models break down in regions of stronger compositional bias or low complexity. While these regions are typically filtered, there is increasing anecdotal evidence of functional roles. This motivates an exploration of more complex sequence models and application-specific approaches for the investigation of biased regions. Results Traditional Markov-chains and application-specific regression models are compared using the example of predicting runs of single amino acids, a particularly simple class of biased regions. Cross-fold validation experiments reveal that the alternative regression models capture the multi-variate trends well, despite their low dimensionality and in contrast even to higher-order Markov-predictors. We show how the significance of unusual observations can be computed for such empirical models. The power of a dedicated model in the detection of biologically interesting signals is then demonstrated in an analysis identifying the unexpected enrichment of contiguous leucine-repeats in signal-peptides. Considering different reference sets, we show how the question examined actually defines what constitutes the 'background'. Results can thus be highly sensitive to the choice of appropriate model training sets. Conversely, the choice of reference data determines the questions that can be investigated in an analysis. Conclusions Using a specific case of studying biased regions as an example, we have demonstrated that the construction of application-specific background models is both necessary and feasible in a challenging sequence analysis situation