Prolongation Loop Algebras for a Solitonic System of Equations

We consider an integrable system of reduced Maxwell-Bloch equations that describes the evolution of an electromagnetic field in a two-level medium that is inhomogeneously broadened. We prove that the relevant Backlund transformation preserves the reality of the n-soliton potentials and establish their pole structure with respect to the broadening parameter. The natural phase space of the model is embedded in an infinite dimensional loop algebra. The dynamical equations of the model are associated to an infinite family of higher order Hamiltonian systems that are in involution. We present the Hamiltonian functions and the Poisson brackets between the extended potentials.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

The modular hierarchy of the Toda lattice

The modular vector field plays an important role in the theory of Poisson manifolds and is intimately connected with the Poisson cohomology of the space. In this paper we investigate its significance in the theory of integrable systems. We illustrate in detail the case of the Toda lattice both in Flaschka and natural coordinates.Comment: 16 pages, 29 references, to appear in Differential Geometry and its application

On ATG4B as Drug Target for Treatment of Solid Tumours-The Knowns and the Unknowns

Autophagy is an evolutionary conserved stress survival pathway that has been shown to play an important role in the initiation, progression, and metastasis of multiple cancers; however, little progress has been made to date in translation of basic research to clinical application. This is partially due to an incomplete understanding of the role of autophagy in the different stages of cancer, and also to an incomplete assessment of potential drug targets in the autophagy pathway. While drug discovery efforts are on-going to target enzymes involved in the initiation phase of the autophagosome, e.g., unc51-like autophagy activating kinase (ULK)1/2, vacuolar protein sorting 34 (Vps34), and autophagy-related (ATG)7, we propose that the cysteine protease ATG4B is a bona fide drug target for the development of anti-cancer treatments. In this review, we highlight some of the recent advances in our understanding of the role of ATG4B in autophagy and its relevance to cancer, and perform a critical evaluation of ATG4B as a druggable cancer targe

A new age in functional genomics using CRISPR/Cas9 in arrayed library screening

CRISPR technology has rapidly changed the face of biological research, such that precise genome editing has now become routine for many labs within several years of its initial development. What makes CRISPR/Cas9 so revolutionary is the ability to target a protein (Cas9) to an exact genomic locus, through designing a specific short complementary nucleotide sequence, that together with a common scaffold sequence, constitute the guide RNA bridging the protein and the DNA. Wild-type Cas9 cleaves both DNA strands at its target sequence, but this protein can also be modified to exert many other functions. For instance, by attaching an activation domain to catalytically inactive Cas9 and targeting a promoter region, it is possible to stimulate the expression of a specific endogenous gene. In principle, any genomic region can be targeted, and recent efforts have successfully generated pooled guide RNA libraries for coding and regulatory regions of human, mouse and Drosophila genomes with high coverage, thus facilitating functional phenotypic screening. In this review, we will highlight recent developments in the area of CRISPR-based functional genomics and discuss potential future directions, with a special focus on mammalian cell systems and arrayed library screening

The Toda lattice is super-integrable

We prove that the classical, non-periodic Toda lattice is super-integrable. In other words, we show that it possesses 2N-1 independent constants of motion, where N is the number of degrees of freedom. The main ingredient of the proof is the use of some special action--angle coordinates introduced by Moser to solve the equations of motion.Comment: 8 page

A reversible phospho-switch mediated by ULK1 regulates the activity of autophagy protease ATG4B

Upon induction of autophagy, the ubiquitin-like protein LC3 is conjugated to phosphatidylethanolamine (PE) on the inner and outer membrane of autophagosomes to allow cargo selection and autophagosome formation. LC3 undergoes two processing steps, the proteolytic cleavage of pro-LC3 and the de-lipidation of LC3-PE from autophagosomes, both executed by the same cysteine protease ATG4. How ATG4 activity is regulated to co-ordinate these events is currently unknown. Here we find that ULK1, a protein kinase activated at the autophagosome formation site, phosphorylates human ATG4B on serine 316. Phosphorylation at this residue results in inhibition of its catalytic activity in vitro and in vivo. On the other hand, phosphatase PP2A-PP2R3B can remove this inhibitory phosphorylation. We propose that the opposing activities of ULK1-mediated phosphorylation and PP2A-mediated dephosphorylation provide a phospho-switch that regulates the cellular activity of ATG4B to control LC3 processing

A symplectic realization of the Volterra lattice

We examine the multiple Hamiltonian structure and construct a symplectic realization of the Volterra model. We rediscover the hierarchy of invariants, Poisson brackets and master symmetries via the use of a recursion operator. The rational Volterra bracket is obtained using a negative recursion operator.Comment: 8 page

Redundancy of human ATG4 protease isoforms in autophagy and LC3/GABARAP processing revealed in cells

Macroautophagy/autophagy is a cellular degradation pathway that delivers cytoplasmic material to lysosomes via double-membrane organelles called autophagosomes. Lipidation of ubiquitin-like LC3/GABARAP proteins on the autophagosome membrane is important for autophagy. The cysteine protease ATG4 executes 2 LC3/GABARAP processing events: priming of newly synthesized pro-LC3/GABARAP to enable subsequent lipidation, and delipidation/deconjugation of lipidated LC3/GABARAP (the exact purpose of which is unclear in mammals). Four ATG4 isoforms (ATG4A to ATG4D) exist in mammals; however, the functional redundancy of these proteins in cells is poorly understood. Here we show that human HAP1 and HeLa cells lacking ATG4B exhibit a severe but incomplete defect in LC3/GABARAP processing and autophagy. By further genetic depletion of ATG4 isoforms using CRISPR-Cas9 and siRNA we uncover that ATG4A, ATG4C and ATGD all contribute to residual priming activity, which is sufficient to enable lipidation of endogenous GABARAPL1 on autophagic structures. We also demonstrate that expressing high levels of pre-primed LC3B in ATG4-deficient cells can rescue a defect in autophagic degradation of the cargo receptor SQSTM1/p62, suggesting that delipidation by human ATG4 is not essential for autophagosome formation and fusion with lysosomes. Overall, our study provides a comprehensive characterization of ATG4 isoform function during autophagy in human cells

Multiscale expansion and integrability properties of the lattice potential KdV equation

We apply the discrete multiscale expansion to the Lax pair and to the first few symmetries of the lattice potential Korteweg-de Vries equation. From these calculations we show that, like the lowest order secularity conditions give a nonlinear Schroedinger equation, the Lax pair gives at the same order the Zakharov and Shabat spectral problem and the symmetries the hierarchy of point and generalized symmetries of the nonlinear Schroedinger equation.Comment: 10 pages, contribution to the proceedings of the NEEDS 2007 Conferenc

Multiscale reduction of discrete nonlinear Schroedinger equations

We use a discrete multiscale analysis to study the asymptotic integrability of differential-difference equations. In particular, we show that multiscale perturbation techniques provide an analytic tool to derive necessary integrability conditions for two well-known discretizations of the nonlinear Schroedinger equation.Comment: 12 page