The central theme of the dissertation is to understand physical principles underlying the dynamical self-organization of the genome in the cell across different length and time scales. The inherently multiscale and out-of equilibrium nature of chromatin dynamics and compaction in the strongly heterogeneous nucleus has been addressed in this work. The large-scale organization of chromatin – the functional form of DNA – is critical to transcription during which ATP-powered enzymes such as RNA polymerases must physically access specific genes within a tightly-packed, micron-scale nucleus. In differentiated cell nuclei, two major chromatin compartments -- heterochromatin and euchromatin -- are spatially segregated, with transcriptionally active euchromatin loosely packed while mostly silent genes are compacted into heterochromatin. Through the lens of an active polymer model of chromosomes using Brownian dynamics simulations based on a boundary integral formulation, along with a kernel-independent fast multipole method, we investigate how large heterochromatic regions form and segregate through complex interactions with euchromatin. Our model consists of a system of long flexible bead-spring polymers representing chromosomes, which are confined to a spheroidal nucleus and composed of alternating segments of active euchromatin and silent heterochromatin immersed in a viscous solvent. Our simulations are applied to analyze the role of euchromatic activity on nuclear dynamics and pattern formation, and demonstrate that active stresses and resulting nucleoplasmic flows serve to enhance heterochromatin segregation and compaction by promoting interactions and subsequent crosslinking between distant genomic segments.

Moving to nano-scale organization, we also analyzed the dynamics of DNA loop extrusion in the presence of the active motor proteins known as structural maintenance of chromosomes (SMC) complexes, of which condensin and cohesin are two important examples. These complexes have been observed to translocate along chromatin in a force-dependent directed manner to form loops via a process known as "loop extrusion" using the energy obtained from the hydrolysis of ATP. In this work, we elucidated the role played by hydrodynamic interactions, the active response of the motors and chain mechanics in assisting loop extrusion under different parameter regimes. The model accounts for the structural features and dynamics of the motor proteins. Furthermore, studies with active polymer in an unconfined environment elucidated the role of activity and hydrodynamics in the self induced coil-stretch transition. This `active coil–stretch transition' is reminiscent of the transition exhibited by passive flexible polymers in externally applied extensional flows, but is internally driven by dipolar activity in the present case. Discussions of the simulation findings are complemented with a simple kinetic model for an active trimer. Our modeling efforts complemented and informed some of the experimental observations providing deeper insight into the mechanisms behind the genome organization.