Changes in the geometry and topology of self-assembled membranes underlie diverse processes across cellular biology and engineering. Similar to lipid bilayers, monolayer colloidal membranes studied by the Sharma (IISc Bangalore) and Dogic (UCSB) Labs have in-plane fluid-like dynamics and out-of-plane bending elasticity, but their open edges and micron length scale provide a tractable system to study the equilibrium energetics and dynamic pathways of membrane assembly and reconfiguration. First, we discuss how doping colloidal membranes with short miscible rods transforms disk-shaped membranes into saddle-shaped minimal surfaces with complex edge structures. Theoretical modeling demonstrates that their formation is driven by increasing positive Gaussian modulus, which in turn is controlled by the fraction of short rods. Further coalescence of saddle-shaped surfaces leads to exotic topologically distinct structures, including shapes similar to catenoids, tri-noids, four-noids, and higher order structures. We then mathematically explore the mechanics of these catenoid-like structures subject to an external axial force and elucidate their intimate connection to two problems whose solutions date back to Euler: the shape of an area-minimizing soap film and the buckling of a slender rod under compression. A perturbation theory argument directly relates the tensions of membranes to the stability properties of minimal surfaces. We also investigate the effects of including a Gaussian curvature modulus, which, for small enough membranes, causes the axial force to diverge as the ring separation approaches its maximal value.

Self-organization is a universal concept spanning numerous disciplines including mathematics, physics and biology. Chromosomes are self-organizing polymers that fold into orderly, hierarchical and yet dynamic structures. In the past decade, advances in experimental biology have provided a means to reveal information about chromosome connectivity, allowing us to directly use this information from experiments to generate 3D models of individual genes, chromosomes and even genomes. In this talk I will present a novel data-driven modeling approach and discuss a number of possibilities that this method holds. I will discuss a detailed study of the time-evolution of X chromosome inactivation, highlighting both global and local properties of chromosomes that result in topology-driven dynamical arrest and present and characterize a novel type of motion we discovered in knots that may have applications to nanoscale materials and machines.

Filaments (slender, microscopic elastic bodies) are prevalent in biological and industrial settings. In the biological case, the filaments are often active, in that they are driven internally by motor proteins, with the prime examples being cilia and flagella. For cilia in particular, which can appear in dense arrays, their resulting motions are coupled through the surrounding fluid, as well as through surfaces to which they are attached. In this talk, I present numerical simulations exploring the coordinated motion of active filaments and how it depends on the driving force, density of filaments, as well as the attached surface. In particular, we find that when the surface is spherical, its topology introduces local defects in coordinated motion which can then feedback and alter the global state. This is particularly true when the surface is not held fixed and is free to move in the surrounding fluid. These simulations take advantage of a computational framework we developed for fully 3D filament motion that combines unit quaternions, implicit geometric time integration, quasi-Newton methods, and fast, matrix-free methods for hydrodynamic interactions and it will also be presented.

Rigidity is the ability of a system to resist imposed stresses before ultimately undergoing failure. However, disordered materials often contain both rigid and floppy subregions that complicate the utility of taking system-wide averages. I will talk about 3 frameworks capable of connecting the internal structure of disordered materials to their rigidity and/or failure under loading, and describe how my collaborators and I have applied these frameworks to laboratory data on laser-cut lattices and idealized granular materials. These are, in order of increasing physics content: (1) centrality within an adjacency matrix describing its connectivity, (2) Maxwell constraint counting on the full network of frictional contact forces, and (3) the vibrational modes of a synthetic dynamical matrix (Hessian). The first two rely primarily on topology, and the second two contrast the utility of considering interparticle forces (Coulomb failure) vs. the energy landscape. All three methods, while successfully elucidating the origins of rigidity and brittle vs. ductile failure, also provide interesting counterpoints regarding how much information is enough to make predictions.

The dream of programmable matter is to create materials whose physical properties (shape, moduli, response to perturbations, etc.) can be changed on the fly. For many years, my group has been thinking about how to program flat sheets that fold up into three dimensional shapes, most recently by exploiting the principles of origami design. Unfortunately, a combinatorial explosion of folding pathways makes robust folding particularly challenging. In this talk, I will discuss how this pluripotency arises from the topology of the configuration space. This suggests a broader understanding of a larger class of materials spanning from folding forms to spring networks to mechanical structures that perform computational logic.

Clock networks containing the same central architectures may vary drastically in their potential to oscillate, raising the question of what controls robustness, one of the essential functions of an oscillator. We computationally generate an atlas of oscillators and found that, while core topologies are critical for oscillations, local structures substantially modulate the degree of robustness. Strikingly, two local structures, incoherent and coherent inputs, can modify a core topology to promote and attenuate its robustness, additively. The findings underscore the importance of local modifications to the performance of the whole network. It may explain why auxiliary structures not required for oscillations are evolutionary conserved. We also extend this computational framework to search hidden network motifs for other clock functions, such as tunability that relates to the capabilities of a clock to adjust timing to external cues. Experimentally, we developed an artificial cell system in water-in-oil microemulsions, within which we reconstitute mitotic cell cycles that can perform self-sustained oscillations for 30 to 40 cycles over multiple days. The oscillation profiles, such as period, amplitude, and shape, can be quantitatively varied with the concentrations of clock regulators, energy levels, droplet sizes, and circuit design. Such innate flexibility makes it crucial to studying clock functions of tunability and stochasticity at the single-cell level. Combined with a pressure-driven multi-channel tuning setup and long-term time-lapse fluorescence microscopy, this system enables a high-throughput exploration in multi-dimension continuous parameter space and single-cell analysis of the clock dynamics and functions. We integrate this experimental platform with mathematical modeling to elucidate the topology-function relation of biological clocks. With FRET and optogenetics, we also investigate spatiotemporal cell-cycle dynamics in both homogeneous and heterogeneous microenvironments by reconstructing subcellular compartments.