Analyzing the dependence structure of microarray data: a copula–based approach

The main aim of this Ph.D. dissertation is the study of clustering dependent data by means of copula functions with particular emphasis on microarray data. Copula functions are a popular multivariate modeling tool in each field where the multivariate dependence is of great interest and their use in clustering has not been still investigated. The first part of this work contains the review of the literature of clustering methods, copula functions and microarray experiments. The attention focuses on the K–means (Hartigan, 1975; Hartigan and Wong, 1979), the hierarchical (Everitt, 1974) and the model–based (Fraley and Raftery, 1998, 1999, 2000, 2007) clustering techniques because their performance is compared. Then, the probabilistic interpretation of the Sklar’s theorem (Sklar’s, 1959), the estimation methods for copulas like the Inference for Margins (Joe and Xu, 1996) and the Archimedean and Elliptical copula families are presented. In the end, applications of clustering methods and copulas to the genetic and microarray experiments are highlighted. The second part contains the original contribution proposed. A simulation study is performed in order to evaluate the performance of the K–means and the hierarchical bottom–up clustering methods in identifying clusters according to the dependence structure of the data generating process. Different simulations are performed by varying different conditions (e.g., the kind of margins (distinct, overlapping and nested) and the value of the dependence parameter ) and the results are evaluated by means of different measures of performance. In light of the simulation results and of the limits of the two investigated clustering methods, a new clustering algorithm based on copula functions (‘CoClust’ in brief) is proposed. The basic idea, the iterative procedure of the CoClust and the description of the written R functions with their output are given. The CoClust algorithm is tested on simulated data (by varying the number of clusters, the copula models, the dependence parameter value and the degree of overlap of margins) and is compared with the performance of model–based clustering by using different measures of performance, like the percentage of well–identified number of clusters and the not rejection percentage of H0 on . It is shown that the CoClust algorithm allows to overcome all observed limits of the other investigated clustering techniques and is able to identify clusters according to the dependence structure of the data independently of the degree of overlap of margins and the strength of the dependence. The CoClust uses a criterion based on the maximized log–likelihood function of the copula and can virtually account for any possible dependence relationship between observations. Many peculiar characteristics are shown for the CoClust, e.g. its capability of identifying the true number of clusters and the fact that it does not require a starting classification. Finally, the CoClust algorithm is applied to the real microarray data of Hedenfalk et al. (2001) both to the gene expressions observed in three different cancer samples and to the columns (tumor samples) of the whole data matrix.

Bioconjugation and synthetic approach towards enantioenriched gem-difluoromethylene compounds through carbenium ions

Bioconjugation of peptides and asymmetric synthesis of gem-difluoromethylene compounds are areas of the modern organic chemistry for which mild and selective methods continue to be developed. This thesis reports new methodologies for these two areas based on the use of stabilized carbenium ions. The reaction that makes the bioconjugation of peptides possible takes place via the direct nucleophilic substitution of alcohols and is driven by the spontaneous formation of stabilized carbenium ions in water. By reacting with the thiol group of cysteine in very mild conditions and with a high selectivity, these carbenium ions allow the site-specific ligation of polypeptides containing cysteine and their covalent derivatization with functionalized probes. The ligation of the indole ring of tryptophan, an emerging target in bioconjugation, is also shown and takes place in the same conditions. The second area investigated is the challenging access to optically active gem-difluoromethylene compounds. We describe a methodology relying on the synthesis of enantioenriched 1,3-benzodithioles intermediates that are shown to be precursors of the corresponding gem-difluoromethylene analogues by oxidative desulfurization-fluorination. This synthesis takes advantage of the highly enantioselective organocatalytic α-alkylation of aldehydes with the benzodithiolylium ion and of the wide possibilities of synthetic transformations offered by the 1,3-benzodithiole group. This approach allows the asymmetric access to complex gem-difluoromethylene compounds through a late-stage fluorination step, thus avoiding the use of fluorinated building blocks.