Flexibility properties in Complex Analysis and Affine Algebraic Geometry

In the last decades affine algebraic varieties and Stein manifolds with big (infinite-dimensional) automorphism groups have been intensively studied. Several notions expressing that the automorphisms group is big have been proposed. All of them imply that the manifold in question is an Oka-Forstneri\v{c} manifold. This important notion has also recently merged from the intensive studies around the homotopy principle in Complex Analysis. This homotopy principle, which goes back to the 1930's, has had an enormous impact on the development of the area of Several Complex Variables and the number of its applications is constantly growing. In this overview article we present 3 classes of properties: 1. density property, 2. flexibility 3. Oka-Forstneri\v{c}. For each class we give the relevant definitions, its most significant features and explain the known implications between all these properties. Many difficult mathematical problems could be solved by applying the developed theory, we indicate some of the most spectacular ones.Comment: thanks added, minor correction

Affine modifications and affine hypersurfaces with a very transitive automorphism group

We study a kind of modification of an affine domain which produces another affine domain. First appeared in passing in the basic paper of O. Zariski (1942), it was further considered by E.D. Davis (1967). The first named author applied its geometric counterpart to construct contractible smooth affine varieties non-isomorphic to Euclidean spaces. Here we provide certain conditions which guarantee preservation of the topology under a modification. As an application, we show that the group of biregular automorphisms of the affine hypersurface $X \subset C^{k+2}$ given by the equation $uv=p(x_1,...,x_k)$ where $p \in C[x_1,...,x_k],$ acts $m-$transitively on the smooth part reg$X$ of $X$ for any $m \in N.$ We present examples of such hypersurfaces diffeomorphic to Euclidean spaces.Comment: 39 Pages, LaTeX; a revised version with minor changes and correction

High dietary folate in pregnant mice leads to pseudo-MTHFR deficiency and altered methyl metabolism, with embryonic growth delay and short-term memory impairment in offspring

Methylenetetrahydrofolate reductase (MTHFR) generates methyltetrahydrofolate for methylation reactions. Severe MTHFR deficiency results in homocystinuria and neurologic impairment. Mild MTHFR deficiency (677C > T polymorphism) increases risk for complex traits, including neuropsychiatric disorders. Although low dietary folate impacts brain development, recent concerns have focused on high folate intake following food fortification and increased vitamin use. Our goal was to determine whether high dietary folate during pregnancy affects brain development in murine offspring. Female mice were placed on control diet (CD) or folic acid-supplemented diet (FASD) throughout mating, pregnancy and lactation. Three-week-old male pups were evaluated for motor and cognitive function. Tissues from E17.5 embryos, pups and dams were collected for choline/methyl metabolite measurements, immunoblotting or gene expression of relevant enzymes. Brains were examined for morphology of hippocampus and cortex. Pups of FASD mothers displayed short-term memory impairment, decreased hippocampal size and decreased thickness of the dentate gyrus. MTHFR protein levels were reduced in FASD pup livers, with lower concentrations of phosphocholine and glycerophosphocholine in liver and hippocampus, respectively. FASD pup brains showed evidence of altered acetylcholine availability and Dnmt3a mRNA was reduced in cortex and hippocampus. E17.5 embryos and placentas from FASD dams were smaller. MTHFR protein and mRNA were reduced in embryonic liver, with lower concentrations of choline, betaine and phosphocholine. Embryonic brain displayed altered development of cortical layers. In summary, high folate intake during pregnancy leads to pseudo-MTHFR deficiency, disturbed choline/methyl metabolism, embryonic growth delay and memory impairment in offspring. These findings highlight the unintended negative consequences of supplemental folic acid

Role of Myotonic Dystrophy Protein Kinase (DMPK) in Glucose Homeostasis and Muscle Insulin Action

Myotonic dystrophy 1 (DM1) is caused by a CTG expansion in the 3′-unstranslated region of the DMPK gene, which encodes a serine/threonine protein kinase. One of the common clinical features of DM1 patients is insulin resistance, which has been associated with a pathogenic effect of the repeat expansions. Here we show that DMPK itself is a positive modulator of insulin action. DMPK-deficient (dmpk−/−) mice exhibit impaired insulin signaling in muscle tissues but not in adipocytes and liver, tissues in which DMPK is not expressed. Dmpk−/− mice display metabolic derangements such as abnormal glucose tolerance, reduced glucose uptake and impaired insulin-dependent GLUT4 trafficking in muscle. Using DMPK mutants, we show that DMPK is required for a correct intracellular trafficking of insulin and IGF-1 receptors, providing a mechanism to explain the molecular and metabolic phenotype of dmpk−/− mice. Taken together, these findings indicate that reduced DMPK expression may directly influence the onset of insulin-resistance in DM1 patients and point to dmpk as a new candidate gene for susceptibility to type 2-diabetes

Active wetting of epithelial tissues

Development, regeneration and cancer involve drastic transitions in tissue morphology. In analogy with the behavior of inert fluids, some of these transitions have been interpreted as wetting transitions. The validity and scope of this analogy are unclear, however, because the active cellular forces that drive tissue wetting have been neither measured nor theoretically accounted for. Here we show that the transition between 2D epithelial monolayers and 3D spheroidal aggregates can be understood as an active wetting transition whose physics differs fundamentally from that of passive wetting phenomena. By combining an active polar fluid model with measurements of physical forces as a function of tissue size, contractility, cell-cell and cell-substrate adhesion, and substrate stiffness, we show that the wetting transition results from the competition between traction forces and contractile intercellular stresses. This competition defines a new intrinsic lengthscale that gives rise to a critical size for the wetting transition in tissues, a striking feature that has no counterpart in classical wetting. Finally, we show that active shape fluctuations are dynamically amplified during tissue dewetting. Overall, we conclude that tissue spreading constitutes a prominent example of active wetting --- a novel physical scenario that may explain morphological transitions during tissue morphogenesis and tumor progression

Holomorphic automorphisms of Danielewski surfaces II -- structure of the overshear group

We apply Nevanlinna theory for algebraic varieties to Danielewski surfaces and investigate their group of holomorphic automorphisms. Our main result states that the overshear group which is known to be dense in the identity component of the holomorphic automorphism group, is a free amalgamated product.Comment: 24 page

Compton scattering beyond the impulse approximation

We treat the non-relativistic Compton scattering process in which an incoming photon scatters from an N-electron many-body state to yield an outgoing photon and a recoil electron, without invoking the commonly used frameworks of either the impulse approximation (IA) or the independent particle model (IPM). An expression for the associated triple differential scattering cross section is obtained in terms of Dyson orbitals, which give the overlap amplitudes between the N-electron initial state and the (N-1) electron singly ionized quantum states of the target. We show how in the high energy transfer regime, one can recover from our general formalism the standard IA based formula for the cross section which involves the ground state electron momentum density (EMD) of the initial state. Our formalism will permit the analysis and interpretation of electronic transitions in correlated electron systems via inelastic x-ray scattering (IXS) spectroscopy beyond the constraints of the IA and the IPM.Comment: 7 pages, 1 figur

Noncovalent Interactions of Hydrated DNA and RNA Mapped by 2D-IR Spectroscopy

Biomolecules couple to their aqueous environment through a variety of noncovalent interactions. Local structures at the surface of DNA and RNA are frequently determined by hydrogen bonds with water molecules, complemented by non-specific electrostatic and many-body interactions. Structural fluctuations of the water shell result in fluctuating Coulomb forces on polar and/or ionic groups of the biomolecular structure and in a breaking and reformation of hydrogen bonds. Two-dimensional infrared (2D-IR) spectroscopy of vibrational modes of DNA and RNA gives insight into local hydration geometries, elementary molecular dynamics, and the mechanisms behind them. In this chapter, recent results from 2D-IR spectroscopy of native and artificial DNA and RNA are presented, together with theoretical calculations of molecular couplings and molecular dynamics simulations. Backbone vibrations of DNA and RNA are established as sensitive noninvasive probes of the complex behavior of hydrated helices. The results reveal the femtosecond fluctuation dynamics of the water shell, the short-range character of Coulomb interactions, and the strength and fluctuation amplitudes of interfacial electric fields.Comment: To appear as Chapter 8 of Springer Series in Optical Sciences: Coherent Multidimensional Spectroscopy -- Editors: Cho, Minhaeng (Ed.), 201