Some Relationships between the Analogs of Euler Numbers and Polynomials

We construct new twisted Euler polynomials and numbers. We also study the generating functions of the twisted Euler numbers and polynomials associated with their interpolation functions. Next we construct twisted Euler zeta function, twisted Hurwitz zeta function, twisted Dirichlet l-Euler numbers and twisted Euler polynomials at non-positive integers, respectively. Furthermore, we find distribution relations of generalized twisted Euler numbers and polynomials. By numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the twisted q-Euler polynomials. Finally, we give a table for the solutions of the twisted q-Euler polynomials

Protein misfolding and dysregulated protein homeostasis in autoinflammatory diseases and beyond.

Cells have a number of mechanisms to maintain protein homeostasis, including proteasome-mediated degradation of ubiquitinated proteins and autophagy, a regulated process of ‘self-eating’ where the contents of entire organelles can be recycled for other uses. The unfolded protein response prevents protein overload in the secretory pathway. In the past decade, it has become clear that these fundamental cellular processes also help contain inflammation though degrading pro-inflammatory protein complexes such as the NLRP3 inflammasome. Signaling pathways such as the UPR can also be co-opted by toll-like receptor and mitochondrial reactive oxygen species signaling to induce inflammatory responses. Mutations that alter key inflammatory proteins, such as NLRP3 or TNFR1, can overcome normal protein homeostasis mechanisms, resulting in autoinflammatory diseases. Conversely, Mendelian defects in the proteasome cause protein accumulation, which can trigger interferon-dependent autoinflammatory disease. In non-Mendelian inflammatory diseases, polymorphisms in genes affecting the UPR or autophagy pathways can contribute to disease, and in diseases not formerly considered inflammatory such as neurodegenerative conditions and type 2 diabetes, there is increasing evidence that cell intrinsic or environmental alterations in protein homeostasis may contribute to pathogenesis