Twists of Plücker coordinates as dimer partition functions

The homogeneous coordinate ring of the Grassmannian Grk,n has a cluster structure defined in terms of planar diagrams known as Postnikov diagrams. The cluster corresponding to such a diagram consists entirely of Plucker coordinates. We introduce a twist map on Grk,n, related to the Berenstein-Fomin-Zelevinsky-twist, and give an explicit Laurent expansion for the twist of an arbitrary Plucker coordinate in terms of the cluster variables associated with a fixed Postnikov diagram. The expansion arises as a (scaled) dimer partition function of a weighted version of the bipartite graph dual to the Postnikov diagram, modified by a boundary condition determined by the Plucker coordinate. We also relate the twist map to a maximal green sequence

Chromatin accessibility dynamics of Chlamydia-infected epithelial cells.

Chlamydia are Gram-negative, obligate intracellular bacterial pathogens responsible for a broad spectrum of human and animal diseases. In humans, Chlamydia trachomatis is the most prevalent bacterial sexually transmitted infection worldwide and is the causative agent of trachoma (infectious blindness) in disadvantaged populations. Over the course of its developmental cycle, Chlamydia extensively remodels its intracellular niche and parasitises the host cell for nutrients, with substantial resulting changes to the host cell transcriptome and proteome. However, little information is available on the impact of chlamydial infection on the host cell epigenome and global gene regulation. Regions of open eukaryotic chromatin correspond to nucleosome-depleted regions, which in turn are associated with regulatory functions and transcription factor binding. We applied formaldehyde-assisted isolation of regulatory elements enrichment followed by sequencing (FAIRE-Seq) to generate temporal chromatin maps of C. trachomatis-infected human epithelial cells in vitro over the chlamydial developmental cycle. We detected both conserved and distinct temporal changes to genome-wide chromatin accessibility associated with C. trachomatis infection. The observed differentially accessible chromatin regions include temporally-enriched sets of transcription factors, which may help shape the host cell response to infection. These regions and motifs were linked to genomic features and genes associated with immune responses, re-direction of host cell nutrients, intracellular signalling, cell-cell adhesion, extracellular matrix, metabolism and apoptosis. This work provides another perspective to the complex response to chlamydial infection, and will inform further studies of transcriptional regulation and the epigenome in Chlamydia-infected human cells and tissues

Reflection group presentations arising from cluster algebras

We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system

A category of wide subcategories

An algebra is said to be τ-tilting finite provided it has only a finite number of τ-rigid objects up to isomorphism. To each such algebra, we associate a category whose objects are the wide subcategories of its category of finite dimensional modules, and whose morphisms are indexed by support τ-tilting pairs

Frieze patterns for punctured discs

We construct frieze patterns of type D N with entries which are numbers of matchings between vertices and triangles of corresponding triangulations of a punctured disc. For triangulations corresponding to orientations of the Dynkin diagram of type D N , we show that the numbers in the pattern can be interpreted as specialisations of cluster variables in the corresponding Fomin-Zelevinsky cluster algebra. This is generalised to arbitrary triangulations in an appendix by Hugh Thomas

Nearly Morita equivalences and rigid objects

If T and T′ are two cluster-tilting objects of an acyclic cluster category related by a mutation, their endomorphism algebras are nearly-Morita equivalent (Buan et al., Cluster-tilted algebras, Trans. Amer. Math. Soc. 359(1) (2007), 323–332 (electronic)), that is, their module categories are equivalent “up to a simple module”. This result has been generalised by D. Yang, using a result of Plamondon, to any simple mutation of maximal rigid objects in a 2-Calabi–Yau triangulated category. In this paper, we investigate the more general case of any mutation of a (non-necessarily maximal) rigid object in a triangulated category with a Serre functor. In that setup, the endomorphism algebras might not be nearly-Morita equivalent and we obtain a weaker property that we call pseudo-Morita equivalence. Inspired by Buan and Marsh (From triangulated categories to module categories via localization II: calculus of fractions, J. Lond. Math. Soc. (2) 86(1) (2012), 152–170; From triangulated categories to module categories via localisation, Trans. Amer. Math. Soc. 365(6) (2013), 2845–2861), we also describe our result in terms of localizations

Control of Single Molecule Fluorescence Dynamics by Stimulated Emission Depletion

The feasibility of manipulating the single molecule absorption-emission cycle using picosecond stimulated emission depletion (STED) is investigated using a stochastic computer simulation. In the simulation the molecule is subjected to repeated excitation and depletion events using time delayed pairs of excitation (PUMP) and depletion (DUMP) pulses derived from a high repetition rate pulsed laser system. The model is used to demonstrate that a significant and even substantial reduction in the occurrence of 'dark states' in the fluorescence emission can be achieved using stimulated emission depletion. Variation in the PUMP-DUMP window allows precise control of the fluorescence yield with substantial increases in the fluorescence intensity observed at early PUMP-DUMP delays

Stimulated emission depletion following two photon excitation

The technique of stimulated emission depletion of fluorescence (STED) from a two photon excited molecular population is demonstrated in the S, excited state of fluorescein in ethylene glycol and methanol. Two photon excitation (pump) is achieved using the partial output of a regeneratively amplified Ti:Sapphire laser in conjunction with an optical parametric amplifier whose tuneable output provides a synchronous depletion (dump) pulse. Time resolved fluorescence intensity and anisotropy measurements of the fluorescein emission are made using picosecond time-correlated single photon counting. Pump-dump time delayed fluorescence intensity measurements are used to characterise the response of the system and to provide additional data on saturation dynamics of the dump transition. Two photon STED is modelled using both approximate analytical techniques in the weak dump limit and by numerical solutions to the appropriate rate equations. The latter are used to fit experimental data from which it is possible to determine the cross-section for the stimulated transition and lifetime of the upper vibrational levels of the ground state

Investigating State Restriction in Fluorescent Protein FRET Using Time-Resolved Fluorescence and Anisotropy

Most fluorescent proteins exhibit multiexponential fluorescence decays, indicating a heterogeneous excited state population. FRET between fluorescent proteins should therefore involve multiple energy transfer pathways. We recently demonstrated the FRET pathways between EGFP and mCherry (mC), upon the dimerization of 3-phosphoinositide dependent protein kinase 1 (PDK1), to be highly restricted. A mechanism for FRET restriction based on a highly unfavorable κ(2) orientation factor arising from differences in donor-acceptor transition dipole moment angles in a far from coplanar and near static interaction geometry was proposed. Here this is tested via FRET to mC arising from the association of glutathione (GSH) and glutathione S-transferase (GST) with an intrinsically homogeneous and more mobile donor Oregon Green 488 (OG). A new analysis of the acceptor window intensity, based on the turnover point of the sensitized fluorescence, is combined with donor window intensity and anisotropy measurements which show that unrestricted FRET to mC takes place. However, a long-lived anisotropy decay component in the donor window reveals a GST-GSH population in which FRET does not occur, explaining previous discrepancies between quantitative FRET measurements of GST-GSH association and their accepted values. This reinforces the importance of the local donor-acceptor environment in mediating energy transfer and the need to perform spectrally resolved intensity and anisotropy decay measurements in the accurate quantification of fluorescent protein FRET

Polarised stimulated emission depletion studies of two-photon excited states

Stimulated emission depletion (STED) population and polarisation dynamics following two-photon excitation are investigated for rhodamine 6G in ethylene glycol. Time resolved fluorescence intensity and polarisation measurements were made using picosecond time-correlated single photon counting (TCSPC). Cross-sections for the stimulated transition were measured between 614nm (2.32 X 10(-16) cm(2)) and 663.5nm (6.05 X 10(-17) cm(2)), ground state vibrational lifetimes were found to vary between 314fs and 467fs. A collinear (180degrees) excitation-detection geometry was employed to investigate re-polarisation of the excited state array yielding fluorescence anisotropies above the two-photon limit. The circumvention of single-photon selection rules is demonstrated allowing the measurement of higher order parameters and correlation functions that are wholly inaccessible to 'conventional' (spontaneous) time resolved fluorescence techniques