Algebraic tools in phylogenomics.

En aquesta tesi interdisciplinar desenvolupem eines algebraiques per a problemes en filogenètica i genòmica. Per estudiar l'evolució molecular de les espècies sovint s'usen models evolutius estocàstics. L'evolució es representa en un arbre (anomenat filogenètic) on les espècies actuals corresponen a fulles de l'arbre i els nodes interiors corresponen a ancestres comuns a elles. La longitud d'una branca de l'arbre representa la quantitat de mutacions que han ocorregut entre les dues espècies adjacents a la branca. Llavors l'evolució de seqüències d'ADN en aquestes espècies es modelitza amb un procés Markov ocult al llarg de l'arbre. Si el procés de Markov se suposa a temps continu, normalment s'assumeix que també és homogeni i, en tal cas, els paràmetres del model són les entrades d'una raó de mutació instantània i les longituds de les branques. Si el procés de Markov és a temps discret, llavors els paràmetres del model són les probabilitats condicionades de substitució de nucleòtids al llarg de l'arbre i no hi ha cap hipòtesi d'homogeneïtat. Aquests últims són els tipus de models que considerem en aquesta tesi i són, per tant, més generals que els de temps continu. Des d'aquesta perspectiva s'estudien els problemes més bàsics de la filogenètica: donat un conjunt de seqüències d'ADN, com decidim quin és el model evolutiu més adequat? com inferim de forma eficient els paràmetres del model? I fins i tot, tal i com també hem provat en aquesta tesi, és possible que les espècies no hagin evolucionat seguint un sol arbre sinó una mescla d'arbres i llavors cal abordar aquestes preguntes en aquest cas més general. Per a models evolutius a temps continu i homogenis, s'ha proposat solucions diverses a aquestes preguntes al llarg de les últimes dècades. En aquesta tesi resolem aquests dos problemes per a models evolutius a temps discret usant tècniques algebraiques provinents d'àlgebra lineal, teoria de grups, geometria algebraica i estadística algebraica. A més a més, la nostra solució per al primer problema és vàlida també per a mescles filogenètiques. Hem fet tests dels mètodes proposats en aquesta tesi sobre dades simulades i dades reals del projectes ENCODE (Encyclopedia Of DNA Elements). Per tal de provar els nostres mètodes hem donat algoritmes per a generar seqüències evolucionant sota un model a temps discret amb un nombre esperat de mutacions prefixat. I així mateix, hem demostrat que aquests algorismes generen totes les seqüències possibles (per la majoria de models). Els tests sobre dades simulades mostren que els mètodes proposats són molt acurats i els resultats sobre dades reals permeten corroborar hipòtesis prèviament formulades. Tots els mètodes proposats en aquesta tesi han estat implementats per a un nombre arbitrari d'espècies i estan disponibles públicament.In this thesis we develop interdisciplinary algebraic tools for genomic and phylogenetic problems. To study the molecular evolution of species one often uses stochastic evolutionary models. The evolution is represented in a tree (called phylogenetic tree) whose leaves represent current species and whose internal nodes correspond to their common ancestors. The length of a branch of the tree represents the number of mutations that have occurred between the two species adjacent to the branch. Then ,the evolution of DNA sequences in these species is modeled with a hidden Markov process along the tree. If the Markov process is assumed to be continuous in time, it is usually assumed homogeneous as well and, if so, the model parameters are the instantaneous rate of mutation and the lengths of the branches. If the Markov process is discrete in time, then the model parameters are the conditional probabilities of nucleotide substitution along the tree and there is no assumption of homogeneity. The latter are the types of models we consider in this thesis and are therefore more general than the homogeneous continuous ones. From this perspective we study the basic problems of phylogenetics: Given a set of DNA sequences, what is the evolutionary model that best fits the data? how can we efficiently infer the model parameters? Also, as we also checked in this thesis, it is possible that species have not evolved along a single tree but a mixture of trees so that we need to address these questions in this more general case. For continuous-time, homogeneous, evolutionary models, several solutions to these questions have been proposed during the last decades. In this thesis we solve these two problems for discrete-time evolutionary models, using algebraic techniques from linear algebra, group theory, algebraic geometry and algebraic statistics. In addition, our solution to the first problem is also valid for phylogenetic mixtures. We have made tests of the methods proposed in this thesis on simulated and real data from ENCODE Project (Encyclopedia Of DNA Elements). To test our methods, we also provide algorithms to generate sequences evolving under discrete-time models with a given expected number of mutations. Even more, we have proved that these algorithms generate all possible sequences (for most models). Tests on simulated data show that the methods are very accurate and our results on real data confirm hypotheses previously formulated. All the methods in this thesis have been implemented for an arbitrary number of species and are publicly available.Postprint (published version

GenNon-h: Generating multiple sequence alignments on nonhomogeneous phylogenetic trees

Background: A number of software packages are available to generate DNA multiple sequence alignments (MSAs) evolved under continuous-time Markov processes on phylogenetic trees. On the other hand, methods of simulating the DNA MSA directly from the transition matrices do not exist. Moreover, existing software restricts to the time-reversible models and it is not optimized to generate nonhomogeneous data (i.e. placing distinct substitution rates at different lineages). Results: We present the first package designed to generate MSAs evolving under discrete-time Markov processes on phylogenetic trees, directly from probability substitution matrices. Based on the input model and a phylogenetic tree in the Newick format (with branch lengths measured as the expected number of substitutions per site), the algorithm produces DNA alignments of desired length. GenNon-h is publicly available for download. Conclusion: The software presented here is an efficient tool to generate DNA MSAs on a given phylogenetic tree. GenNon-h provides the user with the nonstationary or nonhomogeneous phylogenetic data that is well suited for testing complex biological hypotheses, exploring the limits of the reconstruction algorithms and their robustness to such models.Postprint (published version

Fréchet differentiability of regular locally Lipschitzian functions

AbstractThis paper considers Fréchet differentiability almost everywhere in the sense of category of regular, locally Lipschitzian real-valued functions defined on open subsets of a Banach space. It is first shown that, for separable Banach spaces, Clarke's generalized gradient of such a function is a minimal, convex- and compact-valued, upper semicontinuous multifunction. Using a theorem of Christensen and Kenderov it is then shown that, for separable Asplund spaces, such a function is Fréchet differentiable on a dense Gδ subset of its domain

The space of phylogenetic mixtures for equivariant models

The selection of the most suitable evolutionary model to analyze the given molecular data is usually left to biologist's choice. In his famous book, J Felsenstein suggested that certain linear equations satisfied by the expected probabilities of patterns observed at the leaves of a phylogenetic tree could be used for model selection. It remained open the question regarding whether these equations were enough for characterizing the evolutionary model. Here we prove that, for equivariant models of evolution, the space of distributions satisfying these linear equations coincides with the space of distributions arising from mixtures of trees on a set of taxa. In other words, we prove that an alignment is produced from a mixture of phylogenetic trees under an equivariant evolutionary model if and only if its distribution of column patterns satisfies the linear equations mentioned above. Moreover, for each equivariant model and for any number of taxa, we provide a set of linearly independent equations defining this space of phylogenetic mixtures. This is a powerful tool that has already been successfully used in model selection. We also use the results obtained to study identifiability issues for phylogenetic mixtures.Comment: 28 pages, 1 figure; to appear in Algorithms for Molecular Biolog

Immunoreactivity of the AAA plus chaperone ClpB from Leptospira interrogans with sera from Leptospira-infected animals

Citation: Krajewska, J., Arent, Z., Wieckowski, D., Zolkiewski, M., & Kedzierska-Mieszkowska, S. (2016). Immunoreactivity of the AAA plus chaperone ClpB from Leptospira interrogans with sera from Leptospira-infected animals. Bmc Microbiology, 16, 8. doi:10.1186/s12866-016-0774-8Leptospira interrogans is a spirochaete responsible for leptospirosis in mammals. The molecular mechanisms of the Leptospira virulence remain mostly unknown. Recently, it has been demonstrated that L. interrogans ClpB (ClpB(Li)) is essential for bacterial survival under stressful conditions and also during infection. The aim of this study was to provide further insight into the role of ClpB in L. interrogans and answer the question whether ClpB(Li) as a potential virulence factor may be a target of the humoral immune response during leptospiral infections in mammals. Results: ClpB(Li) consists of 860 amino acid residues with a predicted molecular mass of 96.3 kDa and shows multi-domain organization similar to that of the well-characterized ClpB from Escherichia coli. The amino acid sequence identity between ClpB(Li) and E. coli ClpB is 52 %. The coding sequence of the clpB(Li) gene was cloned and expressed in E. coli BL21(DE3) strain. Immunoreactivity of the recombinant ClpB(Li) protein was assessed with the sera collected from Leptospira-infected animals and uninfected healthy controls. Western blotting and ELISA analysis demonstrated that ClpB(Li) activates the host immune system, as evidenced by an increased level of antibodies against ClpB(Li) in the sera from infected animals, as compared to the control group. Additionally, ClpB(Li) was found in kidney tissues of Leptospira-infected hamsters. Conclusions: ClpB(Li) is both synthesized and immunogenic during the infectious process, further supporting its involvement in the pathogenicity of Leptospira. In addition, the immunological properties of ClpB(Li) point to its potential value as a diagnostic antigen for the detection of leptospirosis