On the algebra of local unitary invariants of pure and mixed quantum states

We study the structure of the inverse limit of the graded algebras of local unitary invariant polynomials using its Hilbert series. For k subsystems, we conjecture that the inverse limit is a free algebra and the number of algebraically independent generators with homogenous degree 2m equals the number of conjugacy classes of index m subgroups in a free group on k-1 generators. Similarly, we conjecture that the inverse limit in the case of k-partite mixed state invariants is free and the number of algebraically independent generators with homogenous degree m equals the number of conjugacy classes of index m subgroups in a free group on k generators. The two conjectures are shown to be equivalent. To illustrate the equivalence, using the representation theory of the unitary groups, we obtain all invariants in the m=2 graded parts and express them in a simple form both in the case of mixed and pure states. The transformation between the two forms is also derived. Analogous invariants of higher degree are also introduced.Comment: 14 pages, no figure

A circular order on edge-coloured trees and RNA m-diagrams

We study a circular order on labelled, m-edge-coloured trees with k vertices, and show that the set of such trees with a fixed circular order is in bijection with the set of RNA m-diagrams of degree k, combinatorial objects which can be regarded as RNA secondary structures of a certain kind. We enumerate these sets and show that the set of trees with a fixed circular order can be characterized as an equivalence class for the transitive closure of an operation which, in the case m=3, arises as an induction in the context of interval exchange transformations.Comment: 15 pages, 7 figures. New title. Shortened version, presenting the results more efficientl

High-resolution analysis of cis-acting regulatory networks at the α-globin locus.

We have combined the circular chromosome conformation capture protocol with high-throughput, genome-wide sequence analysis to characterize the cis-acting regulatory network at a single locus. In contrast to methods which identify large interacting regions (10–1000 kb), the 4C approach provides a comprehensive, high-resolution analysis of a specific locus with the aim of defining, in detail, the cis-regulatory elements controlling a single gene or gene cluster. Using the human α-globin locus as a model, we detected all known local and long-range interactions with this gene cluster. In addition, we identified two interactions with genes located 300 kb (NME4) and 625 kb (FAM173a) from the α-globin cluster

On-microscope staging of live cells reveals changes in the dynamics of transcriptional bursting during differentiation

Determining the mechanisms by which genes are switched on and off during development is a key aim of current biomedical research. Gene transcription has been widely observed to occur in a discontinuous fashion, with short bursts of activity interspersed with periods of inactivity. It is currently not known if or how this dynamic behaviour changes as mammalian cells differentiate. To investigate this, using an on-microscope analysis, we monitored mouse α-globin transcription in live cells throughout erythropoiesis. We find that changes in the overall levels of α-globin transcription are most closely associated with changes in the fraction of time a gene spends in the active transcriptional state. We identify differences in the patterns of transcriptional bursting throughout differentiation, with maximal transcriptional activity occurring in the mid-phase of differentiation. Early in differentiation, we observe increased fluctuation in transcriptional activity whereas at the peak of gene expression, in early erythroblasts, transcription is relatively stable. Later during differentiation as α-globin expression declines, we again observe more variability in transcription within individual cells. We propose that the observed changes in transcriptional behaviour may reflect changes in the stability of active transcriptional compartments as gene expression is regulated during differentiation

Identification and single-cell functional characterization of an endodermally biased pluripotent substate in human embryonic stem cells

Human embryonic stem cells (hESCs) display substantial heterogeneity in gene expression, implying the existence of discrete substates within the stem cell compartment. To determine whether these substates impact fate decisions of hESCs we used a GFP reporter line to investigate the properties of fractions of putative undifferentiated cells defined by their differential expression of the endoderm transcription factor, GATA6, together with the hESC surface marker, SSEA3. By single-cell cloning, we confirmed that substates characterized by expression of GATA6 and SSEA3 include pluripotent stem cells capable of long-term self-renewal. When clonal stem cell colonies were formed from GATA6-positive and GATA6-negative cells, more of those derived from GATA6-positive cells contained spontaneously differentiated endoderm cells than similar colonies derived from the GATA6-negative cells. We characterized these discrete cellular states using single-cell transcriptomic analysis, identifying a potential role for SOX17 in the establishment of the endoderm-biased stem cell state

Monomiality principle, Sheffer-type polynomials and the normal ordering problem

We solve the boson normal ordering problem for $(q(a^\dag)a+v(a^\dag))^n$ with arbitrary functions $q(x)$ and $v(x)$ and integer $n$, where $a$ and $a^\dag$ are boson annihilation and creation operators, satisfying $[a,a^\dag]=1$. This consequently provides the solution for the exponential $e^{\lambda(q(a^\dag)a+v(a^\dag))}$ generalizing the shift operator. In the course of these considerations we define and explore the monomiality principle and find its representations. We exploit the properties of Sheffer-type polynomials which constitute the inherent structure of this problem. In the end we give some examples illustrating the utility of the method and point out the relation to combinatorial structures.Comment: Presented at the 8'th International School of Theoretical Physics "Symmetry and Structural Properties of Condensed Matter " (SSPCM 2005), Myczkowce, Poland. 13 pages, 31 reference

Hierarchical Dobinski-type relations via substitution and the moment problem

We consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson ([a,a*]=1) monomials of the form exp(x (a*)^r a), r=1,2,..., under the composition of their exponential generating functions (egf). They turn out to be of Sheffer-type. We demonstrate that two key properties of these sequences remain preserved under substitutional composition: (a)the property of being the solution of the Stieltjes moment problem; and (b) the representation of these sequences through infinite series (Dobinski-type relations). We present a number of examples of such composition satisfying properties (a) and (b). We obtain new Dobinski-type formulas and solve the associated moment problem for several hierarchically defined combinatorial families of sequences.Comment: 14 pages, 31 reference

Reactivation of a developmentally silenced embryonic globin gene

The α- and β-globin loci harbor developmentally expressed genes, which are silenced throughout post-natal life. Reactivation of these genes may offer therapeutic approaches for the hemoglobinopathies, the most common single gene disorders. Here, we address mechanisms regulating the embryonically expressed α-like globin, termed ζ-globin. We show that in embryonic erythroid cells, the ζ-gene lies within a ~65 kb sub-TAD (topologically associating domain) of open, acetylated chromatin and interacts with the α-globin super-enhancer. By contrast, in adult erythroid cells, the ζ-gene is packaged within a small (~10 kb) sub-domain of hypoacetylated, facultative heterochromatin within the acetylated sub-TAD and that it no longer interacts with its enhancers. The ζ-gene can be partially re-activated by acetylation and inhibition of histone de-acetylases. In addition to suggesting therapies for severe α-thalassemia, these findings illustrate the general principles by which reactivation of developmental genes may rescue abnormalities arising from mutations in their adult paralogues