Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy

It is shown that the integrable discrete Schwarzian KP (dSKP) equation which constitutes an algebraic superposition formula associated with, for instance, the Schwarzian KP hierarchy, the classical Darboux transformation and quasi-conformal mappings encapsulates nothing but a fundamental theorem of ancient Greek geometry. Thus, it is demonstrated that the connection with Menelaus' theorem and, more generally, Clifford configurations renders the dSKP equation a natural object of inversive geometry on the plane. The geometric and algebraic integrability of dSKP lattices and their reductions to lattices of Menelaus-Darboux, Schwarzian KdV, Schwarzian Boussinesq and Schramm type is discussed. The dSKP and discrete Schwarzian Boussinesq equations are shown to represent discretizations of families of quasi-conformal mappings.Comment: 26 pages, 9 figure

Balancing the dilution and oddity effects: Decisions depend on body size

Background Grouping behaviour, common across the animal kingdom, is known to reduce an individual's risk of predation; particularly through dilution of individual risk and predator confusion (predator inability to single out an individual for attack). Theory predicts greater risk of predation to individuals more conspicuous to predators by difference in appearance from the group (the ‘oddity’ effect). Thus, animals should choose group mates close in appearance to themselves (eg. similar size), whilst also choosing a large group. Methodology and Principal Findings We used the Trinidadian guppy (Poecilia reticulata), a well known model species of group-living freshwater fish, in a series of binary choice trials investigating the outcome of conflict between preferences for large and phenotypically matched groups along a predation risk gradient. We found body-size dependent differences in the resultant social decisions. Large fish preferred shoaling with size-matched individuals, while small fish demonstrated no preference. There was a trend towards reduced preferences for the matched shoal under increased predation risk. Small fish were more active than large fish, moving between shoals more frequently. Activity levels increased as predation risk decreased. We found no effect of unmatched shoal size on preferences or activity. Conclusions and Significance Our results suggest that predation risk and individual body size act together to influence shoaling decisions. Oddity was more important for large than small fish, reducing in importance at higher predation risks. Dilution was potentially of limited importance at these shoal sizes. Activity levels may relate to how much sampling of each shoal was needed by the test fish during decision making. Predation pressure may select for better decision makers to survive to larger size, or that older, larger fish have learned to make shoaling decisions more efficiently, and this, combined with their size relative to shoal-mates, and attractiveness as prey items influences shoaling decisions

Mode of action of DNA-competitive small molecule inhibitors of tyrosyl DNA phosphodiesterase 2

TDP2 is a 5’-tyrosyl DNA phosphodiesterase important for the repair of DNA adducts generated by non-productive (abortive) activity of topoisomerase II. TDP2 facilitates therapeutic resistance to topoisomerase poisons, which are widely used in the treatment of a range of cancer types. Consequently, TDP2 is an interesting target for the development of small molecule inhibitors that could restore sensitivity to topoisomerase-directed therapies. Previous studies identified a class of deazaflavin-based molecules that showed inhibitory activity against TDP2 at therapeutically useful concentrations, but their mode of action was uncertain. We have confirmed that the deazaflavin series inhibits TDP2 enzyme activity in a fluorescence-based assay, suitable for HTS-screening. We have gone on to determine crystal structures of these compounds bound to a ‘humanised’ form of murine TDP2. The structures reveal their novel mode of action as competitive ligands for the binding site of an incoming DNA substrate, and point the way to generating novel and potent inhibitors of TDP2

A small-molecule PI3Kα activator for cardioprotection and neuroregeneration

Harnessing the potential beneficial effects of kinase signalling through the generation of direct kinase activators remains an underexplored area of drug development1,2,3,4,5. This also applies to the PI3K signalling pathway, which has been extensively targeted by inhibitors for conditions with PI3K overactivation, such as cancer and immune dysregulation. Here we report the discovery of UCL-TRO-1938 (referred to as 1938 hereon), a small-molecule activator of the PI3Kα isoform, a crucial effector of growth factor signalling. 1938 allosterically activates PI3Kα through a distinct mechanism by enhancing multiple steps of the PI3Kα catalytic cycle and causes both local and global conformational changes in the PI3Kα structure. This compound is selective for PI3Kα over other PI3K isoforms and multiple protein and lipid kinases. It transiently activates PI3K signalling in all rodent and human cells tested, resulting in cellular responses such as proliferation and neurite outgrowth. In rodent models, acute treatment with 1938 provides cardioprotection from ischaemia–reperfusion injury and, after local administration, enhances nerve regeneration following nerve crush. This study identifies a chemical tool to directly probe the PI3Kα signalling pathway and a new approach to modulate PI3K activity, widening the therapeutic potential of targeting these enzymes through short-term activation for tissue protection and regeneration. Our findings illustrate the potential of activating kinases for therapeutic benefit, a currently largely untapped area of drug development

High-Density Expression of Ca2+-Permeable ASIC1a Channels in NG2 Glia of Rat Hippocampus

NG2 cells, a fourth type of glial cell in the mammalian CNS, undergo reactive changes in response to a wide variety of brain insults. Recent studies have demonstrated that neuronally expressed acid-sensing ion channels (ASICs) are implicated in various neurological disorders including brain ischemia and seizures. Acidosis is a common feature of acute neurological conditions. It is postulated that a drop in pH may be the link between the pathological process and activation of NG2 cells. Such postulate immediately prompts the following questions: Do NG2 cells express ASICs? If so, what are their functional properties and subunit composition? Here, using a combination of electrophysiology, Ca2+ imaging and immunocytochemistry, we present evidence to demonstrate that NG2 cells of the rat hippocampus express high density of Ca2+-permeable ASIC1a channels compared with several types of hippocampal neurons. First, nucleated patch recordings from NG2 cells revealed high density of proton-activated currents. The magnitude of proton-activated current was pH dependent, with a pH for half-maximal activation of 6.3. Second, the current-voltage relationship showed a reversal close to the equilibrium potential for Na+. Third, psalmotoxin 1, a blocker specific for the ASIC1a channel, largely inhibited proton-activated currents. Fourth, Ca2+ imaging showed that activation of proton-activated channels led to an increase of [Ca2+]i. Finally, immunocytochemistry showed co-localization of ASIC1a and NG2 proteins in the hippocampus. Thus the acid chemosensor, the ASIC1a channel, may serve for inducing membrane depolarization and Ca2+ influx, thereby playing a crucial role in the NG2 cell response to injury following ischemia

Ciliopathy is differentially distributed in the brain of a Bardet-Biedl syndrome mouse model

Bardet-Biedl syndrome (BBS) is a genetically heterogeneous inherited human disorder displaying a pleotropic phenotype. Many of the symptoms characterized in the human disease have been reproduced in animal models carrying deletions or knock-in mutations of genes causal for the disorder. Thinning of the cerebral cortex, enlargement of the lateral and third ventricles, and structural changes in cilia are among the pathologies documented in these animal models. Ciliopathy is of particular interest in light of recent studies that have implicated primary neuronal cilia (PNC) in neuronal signal transduction. In the present investigation, we tested the hypothesis that areas of the brain responsible for learning and memory formation would differentially exhibit PNC abnormalities in animals carrying a deletion of the Bbs4 gene (Bbs4-/-). Immunohistochemical localization of adenylyl cyclase-III (ACIII), a marker restricted to PNC, revealed dramatic alterations in PNC morphology and a statistically significant reduction in number of immunopositive cilia in the hippocampus and amygdala of Bbs4-/- mice compared to wild type (WT) littermates. Western blot analysis confirmed the decrease of ACIII levels in the hippocampus and amygdala of Bbs4-/- mice, and electron microscopy demonstrated pathological alterations of PNC in the hippocampus and amygdala. Importantly, no neuronal loss was found within the subregions of amygdala and hippocampus sampled in Bbs4-/- mice and there were no statistically significant alterations of ACIII immunopositive cilia in other areas of the brain not known to contribute to the BBS phenotype. Considered with data documenting a role of cilia in signal transduction these findings support the conclusion that alterations in cilia structure or neurochemical phenotypes may contribute to the cognitive deficits observed in the Bbs4-/- mouse mode. © 2014 Agassandian et al

Missense mutations in the copper transporter gene ATP7A cause X-Linked distal hereditary motor neuropathy

Distal hereditary motor neuropathies comprise a clinically and genetically heterogeneous group of disorders. We recently mapped an X-linked form of this condition to chromosome Xq13.1-q21 in two large unrelated families. The region of genetic linkage included ATP7A, which encodes a copper-transporting P-type ATPase mutated in patients with Menkes disease, a severe infantile-onset neurodegenerative condition. We identified two unique ATP7A missense mutations (p.P1386S and p.T994I) in males with distal motor neuropathy in two families. These molecular alterations impact highly conserved amino acids in the carboxyl half of ATP7A and do not directly involve the copper transporter's known critical functional domains. Studies of p.P1386S revealed normal ATP7A mRNA and protein levels, a defect in ATP7A trafficking, and partial rescue of a S. cerevisiae copper transport knockout. Although ATP7A mutations are typically associated with severe Menkes disease or its milder allelic variant, occipital horn syndrome, we demonstrate here that certain missense mutations at this locus can cause a syndrome restricted to progressive distal motor neuropathy without overt signs of systemic copper deficiency. This previously unrecognized genotype-phenotype correlation suggests an important role of the ATP7A copper transporter in motor-neuron maintenance and function

The Interaction between the First Transmembrane Domain and the Thumb of ASIC1a Is Critical for Its N-Glycosylation and Trafficking

Acid-sensing ion channel-1a (ASIC1a), the primary proton receptor in the brain, contributes to multiple diseases including stroke, epilepsy and multiple sclerosis. Thus, a better understanding of its biogenesis will provide important insights into the regulation of ASIC1a in diseases. Interestingly, ASIC1a contains a large, yet well organized ectodomain, which suggests the hypothesis that correct formation of domain-domain interactions at the extracellular side is a key regulatory step for ASIC1a maturation and trafficking. We tested this hypothesis here by focusing on the interaction between the first transmembrane domain (TM1) and the thumb of ASIC1a, an interaction known to be critical in channel gating. We mutated Tyr71 and Trp287, two key residues involved in the TM1-thumb interaction in mouse ASIC1a, and found that both Y71G and W287G decreased synaptic targeting and surface expression of ASIC1a. These defects were likely due to altered folding; both mutants showed increased resistance to tryptic cleavage, suggesting a change in conformation. Moreover, both mutants lacked the maturation of N-linked glycans through mid to late Golgi. These data suggest that disrupting the interaction between TM1 and thumb alters ASIC1a folding, impedes its glycosylation and reduces its trafficking. Moreover, reducing the culture temperature, an approach commonly used to facilitate protein folding, increased ASIC1a glycosylation, surface expression, current density and slowed the rate of desensitization. These results suggest that correct folding of extracellular ectodomain plays a critical role in ASIC1a biogenesis and function