Cells compartmentalize biochemical processes taking advantage of physical barriers in the form of membranes. Eukaryotes have a wide diversity of membrane-based compartments that can be used in this context, with the main ones being the extracellular membrane, which separates the inside from the outside of the cell, and the nuclear membrane, that separates the nucleus from the cytoplasm. The nuclear membrane not only isolates and protects the DNA and the transcription and replication processes from the other processes that are taking place in the cytoplasm, but also has an active role in the regulation of cellular signaling. The TGF-ß pathway is one of the most important and conserved signaling cascades, and it takes advantage of compartmentalization using a well-tuned balance between import and export rates of active and inactive forms of key proteins. Thus, compartmentalization serves as an additional regulatory mechanism, physically isolating transcription factors from their targets, influencing the dynamics and strength of signal transduction. his contribution focuses on this biophysical layer of regulation. It begins with an in-depth review of the TGF-ß pathway and the main regulatory processes underlying this compartmentalization mechanism. Next, we focus on several contributions that study how this compartmentalization shapes the dynamics and ultimately the cell response, and how this separation can be used as readout of several cellular responses. Finally, we introduce a simplified mathematical approach to study this process derived using a generalized formulation.

This article serves as a preliminary introduction to the design of a new, open-source applied and computational category theory framework, named Categorica, built on top of the Wolfram Language. Categorica allows one to configure and manipulate abstract quivers, categories, groupoids, diagrams, functors and natural transformations, and to perform a vast array of automated abstract algebraic computations using (arbitrary combinations of) the above structures; to manipulate and abstractly reason about arbitrary universal properties, including products, coproducts, pullbacks, pushouts, limits and colimits; and to manipulate, visualize and compute with strict (symmetric) monoidal categories, including full support for automated string diagram rewriting and diagrammatic theorem-proving. In so doing, Categorica combines the capabilities of an abstract computer algebra framework (thus allowing one to compute directly with epimorphisms, monomorphisms, retractions, sections, spans, cospans, fibrations, etc.) with those of a powerful automated theorem-proving system (thus allowing one to convert universal properties and other abstract constructions into (higher-order) equational logic statements that can be reasoned about and proved using standard automated theorem-proving methods, as well as to prove category-theoretic statements directly using purely diagrammatic methods). Internally, Categorica relies upon state-of-the-art graph and hypergraph rewriting algorithms for its automated reasoning capabilities, and is able to convert seamlessly between diagrammatic/combinatorial reasoning on labeled graph representations and symbolic/abstract reasoning on underlying algebraic representations of all category-theoretic concepts. In this first of two articles introducing the design of the framework, we shall focus principally upon its handling of quivers, categories, diagrams, groupoids, functors and natural transformations, including demonstrations of both its algebraic manipulation and theorem-proving capabilities in each case.

This article provides a proof that the Ramanujan's Inequality given by, \begin{align*} \pi(x)^2 < \frac{e x}{\log x} \pi\left(\frac{x}{e}\right) \end{align*} holds unconditionally for every $x\geq \exp(43.5102146)$. In case for an alternate proof of the result stated above, we shall exploit certain estimates involving the Chebyshev Theta Function, $\vartheta(x)$ in order to derive appropriate bounds for $\pi(x)$, which'll lead us to a much improved condition for the inequality proposed by Ramanujan to satisfy unconditionally.