UMI-tools: Modelling sequencing errors in Unique Molecular Identifiers to improve quantification accuracy.

Unique Molecular Identifiers (UMIs) are random oligonucleotide barcodes that are increasingly used in high-throughput sequencing experiments. Through a UMI, identical copies arising from distinct molecules can be distinguished from those arising through PCR-amplification of the same molecule. However, bioinformatic methods to leverage the information from UMIs have yet to be formalised. In particular, sequencing errors in the UMI sequence are often ignored, or else resolved in an ad-hoc manner. We show that errors in the UMI sequence are common and introduce network based methods to account for these errors when identifying PCR duplicates. Using these methods, we demonstrate improved quantification accuracy both under simulated conditions and real iCLIP and single cell RNA-Seq datasets. Reproducibility between iCLIP replicates and single cell RNA Seq clustering are both improved using our proposed network-based method, demonstrating the value of properly accounting for errors in UMIs. These methods are implemented in the open source UMI-tools software package

Contraction of broken symmetries via Kac-Moody formalism

I investigate contractions via Kac-Moody formalism. In particular, I show how the symmetry algebra of the standard 2-D Kepler system, which was identified by Daboul and Slodowy as an infinite-dimensional Kac-Moody loop algebra, and was denoted by ${\mathbb H}_2$, gets reduced by the symmetry breaking term, defined by the Hamiltonian $$H(\beta)= \frac 1 {2m} (p_1^2+p_2^2)- \frac \alpha r - \beta r^{-1/2} \cos ((\phi-\gamma)/2).$$ For this $H (\beta)$ I define two symmetry loop algebras ${\mathfrak L}_{i}(\beta), i=1,2$, by choosing the `basic generators' differently. These ${\mathfrak L}_{i}(\beta)$ can be mapped isomorphically onto subalgebras of ${\mathbb H}_2$, of codimension 2 or 3, revealing the reduction of symmetry. Both factor algebras ${\mathfrak L}_i(\beta)/I_i(E,\beta)$, relative to the corresponding energy-dependent ideals $I_i(E,\beta)$, are isomorphic to ${\mathfrak so}(3)$ and ${\mathfrak so}(2,1)$ for $E0$, respectively, just as for the pure Kepler case. However, they yield two different non-standard contractions as $E \to 0$, namely to the Heisenberg-Weyl algebra ${\mathfrak h}_3={\mathfrak w}_1$ or to an abelian Lie algebra, instead of the Euclidean algebra ${\mathfrak e}(2)$ for the pure Kepler case. The above example suggests a general procedure for defining generalized contractions, and also illustrates the {\em `deformation contraction hysteresis'}, where contraction which involve two contraction parameters can yield different contracted algebras, if the limits are carried out in different order.Comment: 21 pages, 1 figur

Representations of the Generalized Lie Algebra sl(2)_q

We construct finite-dimensional irreducible representations of two quantum algebras related to the generalized Lie algebra \ssll (2)_q introduced by Lyubashenko and the second named author. We consider separately the cases of $q$ generic and $q$ at roots of unity. Some of the representations have no classical analog even for generic $q$. Some of the representations have no analog to the finite-dimensional representations of the quantised enveloping algebra $U_q(sl(2))$, while in those that do there are different matrix elements.Comment: 14 pages, plain-TEX file using input files harvmac.tex, amssym.de

Entanglement and density-functional theory: testing approximations on Hooke's atom

We present two methods of calculating the spatial entanglement of an interacting electron system within the framework of density-functional theory. These methods are tested on the model system of Hooke's atom for which the spatial entanglement can be calculated exactly. We analyse how the strength of the confining potential affects the spatial entanglement and how accurately the methods that we introduced reproduce the exact trends. We also compare the results with the outcomes of standard first-order perturbation methods. The accuracies of energies and densities when using these methods are also considered.Comment: 14 pages with 18 figures; corrected typos, corrected expression for first-order energy in section VI and consequently Fig.13, conclusions and other results unaffecte

Modification of Seurat v4 for the development of a phase assignment tool able to distinguish between G2 and mitotic cells

Single-cell RNA sequencing (scRNAseq) is a rapidly advancing field enabling the characterisation of heterogeneous gene expression profiles within a population. The cell cycle phase is a major contributor to gene expression variance between cells and computational analysis tools have been developed to assign cell cycle phases to cells within scRNAseq datasets. Whilst these tools can be extremely useful, all have the drawback that they classify cells as only G1, S or G2/M. Existing discrete cell phase assignment tools are unable to differentiate between G2 and M and continuous-phase-assignment tools are unable to identify a region corresponding specifically to mitosis in a pseudo-timeline for continuous assignment along the cell cycle. In this study, bulk RNA sequencing was used to identify differentially expressed genes between mitotic and interphase cells isolated based on phospho-histone H3 expression using fluorescence-activated cell sorting. These gene lists were used to develop a methodology which can distinguish G2 and M phase cells in scRNAseq datasets. The phase assignment tools present in Seurat were modified to allow for cell cycle phase assignment of all stages of the cell cycle to identify a mitotic-specific cell population

At what time does a quantum experiment have a result?

This paper provides a general method for defining a generalized quantum observable (or POVM) that supplies properly normalized conditional probabilities for the time of occurrence (i.e., of detection). This method treats the time of occurrence as a probabilistic variable whose value is to be determined by experiment and predicted by the Born rule. This avoids the problematic assumption that a question about the time at which an event occurs must be answered through instantaneous measurements of a projector by an observer, common to both Rovelli (1998) and Oppenheim et al. (2000). I also address the interpretation of experiments purporting to demonstrate the quantum Zeno effect, used by Oppenheim et al. (2000) to justify an inherent uncertainty for measurements of times.Comment: To appear in proceedings of 2015 ETH Zurich Workshop on Time in Physic

Pairwise entanglement in the XX model with a magnetic impurity

For a 3-qubit Heisenberg model in a uniform magnetic field, the pairwise thermal entanglement of any two sites is identical due to the exchange symmetry of sites. In this paper we consider the effect of a non-uniform magnetic field on the Heisenberg model, modeling a magnetic impurity on one site. Since pairwise entanglement is calculated by tracing out one of the three sites, the entanglement clearly depends on which site the impurity is located. When the impurity is located on the site which is traced out, that is, when it acts as an external field of the pair, the entanglement can be enhanced to the maximal value 1; while when the field acts on a site of the pair the corresponding concurrence can only be increased from 1/3 to 2/3.Comment: 9 Pages, 4 EPS figures, LaTeX 2

Z3_3-graded differential geometry of quantum plane

In this work, the Z$_3$-graded differential geometry of the quantum plane is constructed. The corresponding quantum Lie algebra and its Hopf algebra structure are obtained. The dual algebra, i.e. universal enveloping algebra of the quantum plane is explicitly constructed and an isomorphism between the quantum Lie algebra and the dual algebra is given.Comment: 17 page

Three-qubit pure-state canonical forms

In this paper we analyze the canonical forms into which any pure three-qubit state can be cast. The minimal forms, i.e. the ones with the minimal number of product states built from local bases, are also presented and lead to a complete classification of pure three-qubit states. This classification is related to the values of the polynomial invariants under local unitary transformations by a one-to-one correspondence.Comment: REVTEX, 9 pages, 1 figur