Bounded discrete walks

International audienceThis article tackles the enumeration and asymptotics of directed lattice paths (that are isomorphic to unidimensional paths) of bounded height (walks below one wall, or between two walls, for $\textit{any}$ finite set of jumps). Thus, for any lattice paths, we give the generating functions of bridges ("discrete'' Brownian bridges) and reflected bridges ("discrete'' reflected Brownian bridges) of a given height. It is a new success of the "kernel method'' that the generating functions of such walks have some nice expressions as symmetric functions in terms of the roots of the kernel. These formulae also lead to fast algorithms for computing the $n$-th Taylor coefficients of the corresponding generating functions. For a large class of walks, we give the discrete distribution of the height of bridges, and show the convergence to a Rayleigh limit law. For the family of walks consisting of a $-1$ jump and many positive jumps, we give more precise bounds for the speed of convergence. We end our article with a heuristic application to bioinformatics that has a high speed-up relative to previous work

BlastMultAl, a Blast Extension for Similarity Searching with Alignment Graphs

We describe a new method of processing similarity queries of a proteic multiple alignment with a set (database) of protein sequences, or similarity queries of a protein sequence with a set of protein alignments. We use a representation of multiple alignments as alignment-graphs. Comparisons with different classical methods is made. This new method allows the detection of subtle similarities which are not found by the other methods. It has direct applications for similarities querying with the database of protein domains ProDom

Le modèle probabiliste de l'alignement local de deux séquences

Nous décrivons le modèle probabiliste utilisé pour calibrer l'alignement local de deux séquences, quand un schéma de score additif est utilisé. Ce modèle est utilisé en particulier par le logiciel BLAST (Basic Local Alignment Search Tool), qui est employé fréquemment par les biologistes. Asymptotiquement, la probabilité associée au score d'un alignement optimal vérifie une loi des valeurs extrêmes. Nous analysons différents aspects des outils mathématiques (modèles de marche aléatoire, décomposition de Wiener-Hopf, théorie du renouvellement) utilisés dans ce modèle probabiliste

Constructions for Clumps Statistics.

International audienceWe consider a component of the word statistics known as clump; starting from a finite set of words, clumps are maximal overlapping sets of these occurrences. This object has first been studied by Schbath with the aim of counting the number of occurrences of words in random texts. Later work with similar probabilistic approach used the Chen-Stein approximation for a compound Poisson distribution, where the number of clumps follows a law close to Poisson. Presently there is no combinatorial counterpart to this approach, and we fill the gap here. We also provide a construction for the yet unsolved problem of clumps of an arbitrary finite set of words. In contrast with the probabilistic approach which only provides asymptotic results, the combinatorial method provides exact results that are useful when considering short sequences

Constant time estimation of ranking statistics by analytic combinatorics

International audienceWe consider i.i.d. increments (or jumps) X_i that are integers in [-c,...,+d], the partial sums S_j, and the discrete walks (j,S_j) with 1 <= j <= n. Late conditioning by a return of the walk to zero at time n provides discrete bridges that we note B_j with 1<= j <= n. We give in this extended abstract the asymptotic law in the central domain of the height max_{1<= j <= n}B_j of the bridges as n tends to infinity. As expected, this law converges to the Rayleigh law which is the law of the maximum of a standard Brownian bridge. In the case where c=1 (only one negative jump), we provide a full expansion of the asymptotic limit which improves upon the rate of convergence O(log(n)/sqrt(n)) given by Borisov (78) for lattice jumps; this applies in particular to the case where X_i is in {-1,+d}, in which case the expansion is expressible as a function of n, d and of the height of the bridge. Applying this expansion for X_i in {-1,d/c} gives an excellent approximation of the case X_i in {-d,+c} and provides in constant time an indicator used in ranking statistics; this indicator can be used for medical diagnosis and bioinformatics analysis (see Keller et al. (2007)) who compute it in time O(n min(c,d)) by use of dynamical programming)

APPLICATION OF DESIGN SPACE OPTIMIZATION STRATEGY TO THE DEVELOPMENT OF LC METHODS FOR SIMULTANEOUS ANALYSIS OF 18 ANTIRETROVIRAL MEDICINES AND 4 MAJOR EXCIPIENTS USED IN VARIOUS PHARMACEUTICAL FORMULATIONS

peer reviewedtAs one of the world’s most significant public health challenges in low- and middle-income countries,HIV/AIDS deserves to be treated with appropriate medicines, however which are not spared from coun-terfeiting. For that, we developed screening and specific HPLC methods that can analyze 18 antiretroviralmedicines (ARV) and 4 major excipients. Design of experiments and design space methodology wereinitially applied for 15 ARV and the 4 excipients with prediction thanks to Monte Carlo simulations andfocusing on rapidity and affordability thus using short column and low cost organic solvent (methanol)in gradient mode with 10 mM buffer solutions of ammonium hydrogen carbonate. Two other specificmethods dedicated to ARV in liquid and in solid dosage formulations were also predicted and opti-mized. We checked the ability of one method for the analysis of a fixed-dose combination composedby emtricitabine/tenofovir/efavirenz in tablet formulations. Satisfying validation results were obtainedby applying the total error approach taking into account the accuracy profile as decision tool. Then, thevalidated method was applied to test two samples coded A and B, and claimed to contain the tested ARV.Assay results were satisfying only for sample B

Are there clinical variables determining antibiotic prophylaxis-susceptible versus resistant infection in open fractures?

Purpose: In Gustilo grade III open fractures, it remains unknown which demographic or clinical features may be associated with an infection resistant to the administered prophylactic agent, compared to one that is susceptible. Methods: This was a retrospective case-control study on patients hospitalized from 2004 to 2009. Results: We identified 310 patients with Gustilo-III open fractures, 36 (12%) of which became infected after a median of tendays. In 26 (72%) of the episodes the pathogen was susceptible to the prophylactic antibiotic agent prescribed upon admission, while in the other ten it was resistant. All antibiotic prophylaxis was intravenous; the median duration of treatment was threedays and the median delay between trauma and surgery was oneday. In multivariate analysis adjusting for case-mix, only Gustilo-grade-IIIc fractures (vascular lesions) showed tendency to be infected with resistant pathogens (odds ratio 10; 95% confidence interval 1.0-10; p = 0.058). There were no significant differences between cases caused by antibiotic resistant and susceptible pathogen cases in patient's sex, presence of immune suppression, duration and choice of antibiotic prophylaxis, choice of surgical technique or materials, time delay until surgery, use of bone reaming, fracture localization, or presence of compartment syndrome. Conclusion: We were unable to identify any specific clinical parameters associated with infection with antibiotic resistant pathogens in Gustilo-grade III open fractures, other than the severity of the fracture itself. More research is needed to identify patients who might benefit from a broader-spectrum antibiotic prophylaxis

BlastMultAl, a Blast extension for similarity searching with alignment graphs

. We describe a new method of processing similarity queries of a proteic multiple alignment with a set (database) of protein sequences, or similarity queries of a protein sequence with a set of protein alignments. We use a representation of multiple alignments as alignment-graphs. Comparisons with different classical methods is made. This new method allows the detection of subtle similarities which are not found by the other methods. It has direct applications for similarities querying with the database of protein domains ProDom. BlastMultAl: une extension de Blast pour la recherche de similarit&apos;es au moyen de graphes d&apos;alignement. R&apos;esum&apos;e. Nous d&apos;ecrivons une nouvelle m&apos;ethode de recherche de similarit&apos;es d&apos;un alignement multiple de prot&apos;eines avec un ensemble (une base de donn&apos;ees) de s&apos;equences prot&apos;eiques, ou de recherche de similarit&apos;es d&apos;une s&apos;equence prot&apos;eique avec un ensemble d&apos;alignements multiples de prot&apos;eines. Nous comparons cette approche avec des approches classiques..

Average profiles, from tries to suffix-trees

internal profile of tries and of suffix-trees. The binary keys and the strings are built from a Bernoulli source (p, q). We consider the average number pk,P(ν) of internal nodes at depth k of a trie whose number of input keys follows a Poisson law of parameter ν. The Mellin transform of the corresponding bivariate generating function has a major singularity at the origin, which implies a phase reversal for the saturation rate pk,P(ν)/2 k as k reaches the value 2 log(ν)/(log(1/p) + log(1/q)). We prove that the asymptotic average profiles of random tries and suffix-trees are mostly similar, up to second order terms, a fact that has been experimentally observed in Nicodème (2003); the proof follows from comparisons to the profile of tries in the Poisson model