Active matter describes a class of systems that are maintained far from equilibrium by driving forces acting on the constituent particles. Here I will focus on confined active nematics, which exhibit especially rich flow behavior, ranging from structured patterns in space and time to disordered turbulent flows. To understand this behavior, I will take a deterministic dynamical systems approach, beginning with the hydrodynamic equations for the active nematic. This approach reveals that the infinite-dimensional phase space of all possible flow configurations is populated by Exact Coherent Structures (ECS), which are exact solutions of the hydrodynamic equations with distinct and regular spatiotemporal structure; examples include unstable equilibria, periodic orbits, and traveling waves. The ECS are connected by dynamical pathways called invariant manifolds. The main hypothesis in this approach is that turbulence corresponds to a trajectory meandering in the phase space, transitioning between ECS by traveling on the invariant manifolds. Similar approaches have been successful in characterizing high Reynolds number turbulence of passive fluids. Here, I will present the first systematic study of active nematic ECS and their invariant manifolds and discuss their role in characterizing the phenomenon of active turbulence.

Chromosomes are extremely long, active polymers that are spatially organized across multiple scales to promote cellular functions, such as gene transcription and genetic inheritance. During each cell cycle, chromosomes are dramatically compacted as cells divide and dynamically reorganized into less compact, spatiotemporally patterned structures after cell division. These activities are facilitated by DNA/chromatin-binding protein motors called SMC complexes. Each of these motors can perform a unique activity known as “loop extrusion,” in which the motor binds the DNA/chromatin polymer, reels in the polymer fiber, and extrudes it as a loop. Using simulations and theory, I show how loop-extruding motors can collectively compact and spatially organize chromosomes in different scenarios. First, I show that loop-extruding complexes can generate sufficient compaction for cell division, provided that loop-extrusion satisfies stringent physical requirements. Second, while loop-extrusion alone does not uniquely spatially pattern the genome, interactions between SMC complexes and protein “boundary elements” can generate patterns that emerge in the genome after cell division. Intriguingly, these “boundary elements” are not necessarily stationary, which can generate a variety of patterns in the neighborhood of transcriptionally active genes. These predictions, along with supporting experiments, show how SMC complexes and other molecular machinery, such as RNA polymerase, can spatially organize the genome. More generally, this work demonstrates both the versatility of the loop extrusion mechanism for chromosome functional organization and how seemingly subtle microscopic effects can emerge in the spatiotemporal structure of nonequilibrium polymers.