Compartmentalization is ubiquitous in biology, enabling the spatiotemporal regulation of various biochemical processes. Historically, cells have been thought to achieve compartmentalization via membrane-bound bodies known as organelles. However, a new paradigm has recently emerged in which cells can form membraneless compartments called biomolecular condensates. In this thesis, I seek to uncover the physical principles underlying the spatiotemporal regulation of both membrane-bound and membraneless organelles. First, I will discuss the association between microtubules and tau proteins, the dysregulation of which is implicated in neurodegenerative diseases. Through a dialogue between theory and experiments, we show that tau molecules form multilayered condensates via a prewetting-like transition. These condensates promote the recruitment of tau interactors, such as tubulin and other microtubule-associated proteins, thereby allowing for various physiological functions on the microtubule surface. In related work, I will discuss how protein–protein interactions affect transcription factor (TF) target search kinetics. Timely gene expression requires TFs to locate their specific binding sites on DNA. Previous studies have shown that this search is optimized by facilitated diffusion, which combines 3D diffusion with 1D sliding. Here, we show that in the presence of TF–TF interactions, simple 3D diffusion can outperform facilitated diffusion. In line with our results, recent experiments in live cells have shown TFs locating their target sites through 3D diffusion alone. Our results highlight an alternative mechanism cells can use to accelerate target search. In the last part, I focus on membrane-bound organelles. Cells tightly regulate organelle abundances in response to external cues, but the underlying mechanisms remain unclear. We employ a theoretical model that uses time-lapse microscopy datasets to infer the mechanisms regulating organelle biogenesis. Together, this work advances our understanding of intracellular organization across membraneless and membrane-bound organelles.

Ermakov has shown that harmonic oscillator with time dependent frequency can be rescaled to a time independent one along with a nonlinear equation. This analysis can be extended to QHE with time dependent magnetic fields. This allows to bring in Generalized Laughlin wavefunctions. This helps to study dynamics of density fluctuations (GMP) and edgemodes in integer QHE. We will discuss briefly the case of inhomogeneous magnetic fields also and point out interesting connections to Kosambi, Cartan, Chern (KCC) theory.

In this talk, we will first briefly introduce C*-dynamical systems and crossed product C*-algebras. We will then introduce the notion of Rokhlin dimension for group actions and discuss its role in studying the structure of crossed products. Finally, we will present some permanence results showing how suitable regularity properties of the underlying C*-algebra can be inherited by the crossed product under appropriate assumptions on the action.