MotivationWhen gene duplication occurs, one of the copies may become free of selective pressure and evolve at an accelerated pace. This has important consequences on the prediction of orthology relationships, since two orthologous genes separated by divergence after duplication may differ in both sequence and function. In this work, we make the distinction between the primary orthologs, which have not been affected by accelerated mutation rates on their evolutionary path, and the secondary orthologs, which have. Similarity-based prediction methods will tend to miss secondary orthologs, whereas phylogeny-based methods cannot separate primary and secondary orthologs. However, both types of orthology have applications in important areas such as gene function prediction and phylogenetic reconstruction, motivating the need for methods that can distinguish the two types.ResultsWe formalize the notion of divergence after duplication and provide a theoretical basis for the inference of primary and secondary orthologs. We then put these ideas to practice with the Hybrid Prediction of Paralogs and Orthologs (HyPPO) framework, which combines ideas from both similarity and phylogeny approaches. We apply our method to simulated and empirical datasets and show that we achieve superior accuracy in predicting primary orthologs, secondary orthologs and paralogs.Availability and implementationHyPPO is a modular framework with a core developed in Python and is provided with a variety of C++ modules. The source code is available at https://github.com/manuellafond/HyPPO.Supplementary information Supplementary data are available at Bioinformatics online.

Motivation: When gene duplication occurs, one of the copies may become free of selective pressure and evolve at an accelerated pace. This has important consequences on the prediction of orthology relationships, since two orthologous genes separated by divergence after duplication may differ in both sequence and function. In this work, we make the distinction between the primary orthologs, which have not been affected by accelerated mutation rates on their evolutionary path, and the secondary orthologs, which have. Similarity-based prediction methods will tend to miss secondary orthologs, whereas phylogenybased methods cannot separate primary and secondary orthologs. However, both types of orthology have applications in important areas such as gene function prediction and phylogenetic reconstruction, motivating the need for methods that can distinguish the two types. Results: We formalize the notion of divergence after duplication, and provide a theoretical basis for the inference of primary and secondary orthologs. We then put these ideas to practice with the HyPPO (Hybrid Prediction of Paralogs and Orthologs) framework, which combines ideas from both similarity and phylogeny approaches. We apply our method to simulated and empirical datasets, and show that we achieve superior accuracy in predicting primary orthologs, secondary orthologs and paralogs. Availability: HyPPO is a modular framework with a core developed in Python, and is provided with a variety of C++ modules. The source code is available at https://github.com/manuellafond/HyPPO.

We analyze models of genome evolution based on both restricted and unrestricted doublecut-and-join (DCJ) operations. We compare the number of operations along the evolutionary trajectory to the DCJ distance of the genome from its ancestor at each step, and determine at what point they diverge: the process escapes from parsimony. Adapting the method developed by Berestycki and Durret [1], we estimate the number of cycles in the breakpoint graph of a random genome at time t and its ancestral genome by the number of tree components of an Erdös-Rényi random graph constructed from the model of evolution. In both models, the process on a genome of size n is bound to its parsimonious estimate up to t ≈ n/2 steps.

As manifested in Shor's groundbreaking seminal work, quantum mechanics promise the possibility of having substantially more effective computation devices. This is in fact the result of quantum parallelism: the coherent interference pattern between the multitude of superpositions. But the fragility of a quantum state, which on one hand is used to take advantage of the power of entanglement, also can result in undesired interference between the state of the quantum system, carrying or storing the information, with the environment. The problem of maintaining quantum coherence remains one of the most important obstacles in the attempt of exploiting the new possibilities opened up by applications of quantum mechanics in classical computations. Redundancy, which is the main tool in all classical error correcting schemes is no more available in quantum setting, because of the no-cloning theorem, that prevents the duplication of quantum states. The seminal independent work of Shor [32] and Steane [35] gave birth to the current active theory of quantum error correction, which is the subject of this thesis. This thesis is a non-technical introduction to the theory of quantum error correction. We present the mathematical formalism for the theory, covering the main ideas of correcting schemes and techniques, laid in a mathematical framework, avoiding unnecessary computational and physical technicalities. Foremost, I would like to thank my supervisors Dr. Mehrdad Kalantar and Dr. Matthias Neufang for their priceless guidance, support and encouragement provided throughout my graduate studies. I am more than thankful to these wonderful professors for their time and effort devoted during these two years. I would also like to thank Dr. Kennedy for his kind support and help during these years. I also want to thank the School of Mathematics and Statistics and their kind staff. Finally, I would like to thank my lovely parents, my dearest sister and my amazing friends, especially Pegah, for their unconditional love and support. I am very lucky to have these beautiful people during this journey.

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