Second order tangent bundles of infinite dimensional manifolds

The second order tangent bundle $T^{2}M$ of a smooth manifold $M$ consists of the equivalent classes of curves on $M$ that agree up to their acceleration. It is known that in the case of a finite $n$-dimensional manifold $M$, $T^{2}M$ becomes a vector bundle over $M$ if and only if $M$ is endowed with a linear connection. Here we extend this result to $M$ modeled on an arbitrarily chosen Banach space and more generally to those Fr\'{e}chet manifolds which can be obtained as projective limits of Banach manifolds. The result may have application in the study of infinite-dimensional dynamical systems.Comment: 8 page

Isomorphism classes for Banach vector bundle structures of second tangents

On a smooth Banach manifold M, the equivalence classes of curves that agree up to acceleration form the second order tangent bundle T 2M of M. This is a vector bundle in the presence of a linear connection ∇ on M and the corresponding local structure is heavily dependent on the choice of ∇. In this paper we study the extent of this dependence and we prove that it is closely related to the notions of conjugate connections and second order differentials. In particular, the vector bundle structure on T 2 M remains invariant under conjugate connections with respect to diffeomorphisms of M. © 2006 Cambridge Philosophical Society

A generalized second-order frame bundle for Fréchet manifolds

Working within the framework of Fréchet modelled infinite-dimensional manifolds, we propose a generalized notion of second-order frame bundle. We revise in this way the classical notion of bundles of linear frames of order 2, the direct definition and study of which is problematic due to intrinsic difficulties of the space models. However, this new structure keeps all the fundamental characteristics of a frame bundle. It is a principal Fréchet bundle associated (differentially and geometrically) with the corresponding second-order tangent bundle. © 2004 Elsevier B.V. All rights reserved

Conjugate connections and differential equations on infinite dimensional manifolds

On a smooth manifold M, the vector bundle structures of the second order tangent bundle, T^2M, bijectively correspond to linear connections. In this paper we classify such structures for those Frechet manifolds which can be considered as projective limits of Banach manifolds. We investigate also the relation between ordinary differential equations on Frechet spaces and the linear connections on their trivial bundle. Such equations arise in theoretical physics

Structural origins of the functional divergence of human insulin-like growth factor-I and insulin

Human insulin-like growth factors I and II (hIGF-I, hIGF-II) are potent stimulators of cell and growth processes. They display high sequence similarity to both the A and B chains of insulin but contain an additional connecting C-domain, which reflects their secretion without specific packaging or precursor conversion. IGFs also have an extension at the C-terminus known as the D-domain. This paper describes four homologous hIGF-1 structures, obtained from crystals grown in the presence of the detergent SB12, which reveal additional detail in the C- and D-domains. Two different detergent binding modes observed in the crystals may reflect different hIGF-I biological properties such as the interaction with IGF binding proteins and self-aggregation. While the helical core of hIGF-I is very similar to that in insulin, there are distinct differences in the region of hIGF-I corresponding to the insulin B chain C-terminus, residues B25−B30. In hIGF-I, these residues (24−29) and the following C-domain form an extensive loop protruding 20 Å from the core, which results in a substantially different conformation for the receptor binding epitope in hIGF-I compared to insulin. One notable feature of the structures presented here is demonstration of peptide-bond cleavage between Ser35 and Arg36 resulting in an apparent gap between residues 35 and 39. The equivalent region of proinsulin is involved in hormone processing demanding a reassessment of the structural integrity of hIGF-I in relation to its biological function

The NRF2-p97-NRF2 negative feedback loop

p97 is a ubiquitin-targeted ATP-dependent segregase that regulates proteostasis, in addition to a variety of other cellular functions. Previously, we demonstrated that p97 negatively regulates NRF2 by extracting ubiquitylated NRF2 from the KEAP1-CUL3-RBX1 E3 ubiquitin ligase complex, facilitating proteasomal destruction. In the current study, we identified p97 as an NRF2-target gene that contains a functional ARE, indicating the presence of an NRF2-p97-NRF2 negative feedback loop that maintains redox homeostasis. Using CRISPR/Cas9 genome editing, we generated endogenous p97 ARE-mutated BEAS-2B cell lines. These p97 ARE-mutated cell lines exhibit altered expression of p97 and NRF2, as well as a compromised response to NRF2 inducers. Importantly, we also found a positive correlation between NRF2 activation and p97 expression in human cancer patients. Finally, using chronic arsenic-transformed cell lines, we demonstrated a synergistic effect of NRF2 and p97 inhibition in killing cancer cells with high NRF2 and p97 expression. Our study suggests dual upregulation of NRF2 and p97 occurs in certain types of cancers, suggesting that inhibition of both NRF2 and p97 could be a promising treatment strategy for stratified cancer patients