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.

Living organisms have mastered the dynamic control of stresses within sheets to induce shape transformation and locomotion. For instance, the spatiotemporal pattern of action potential in a heart yields a dynamical stress field leading to shape changes and biological function. Such structures inspired the development of theoretical tools and responsive materials alike. Yet, present attempts to mimic their rich dynamics and phenomenology in autonomous synthetic matter are still very limited. In this talk, I will present several complementing innovations toward this goal: novel shaping mechanisms that were overlooked by previous research, new fabrication techniques for programmable matter via 4D printing of gel structures, and most prominently, the first autonomous shape morphing membranes. The dynamical control over the geometry of the material is a prevalent theme in all of these achievements. In particular, the latter system demonstrates localized deformations, induced by a pattern-forming chemical reaction, that prescribe the patterns of curvature, leading to global shape evolution. Together, these developments present a route for modeling and producing fully autonomous soft membranes mimicking some of the locomotive capabilities of living organisms.

Understanding the transport of microorganisms and self-propelled particles in porous media has important consequences in human health as well as for microbial ecology. In this work, we explore models for the dispersion of active particles in both periodic and random porous media. In a first problem, we analyze the long-time transport properties in a dilute system of active Brownian particles swimming in a periodic lattice in the presence of an external flow. Using generalized Taylor dispersion theory, we calculate the mean transport velocity and dispersion dyadic and explain their dependence on flow strength, swimming activity and geometry. In a second approach, we address the case of run-and-tumble particles swimming through unstructured porous media composed of randomly distributed circular pillars. There, we show that the long-time dispersion is described by a universal hindrance function that depends on the medium porosity and ratio of the swimmer run length to the pillar size. An asymptotic expression for the hindrance function is derived in dilute media, and its extension to semi-dilute and dense media is obtained using stochastic simulations. We conclude by discussing the role of hydrodynamic interactions and swimmer concentration effects.

The ability of the brain to encode and store information depends on the plastic nature of the individual synapses. The increase and decrease in synaptic strength, mediated through the structural plasticity of the spine, are important for learning, memory, and cognitive function. Dendritic spines are small structures that contain the synapse. They come in a variety of shapes (stubby, thin, or mushroom-shaped) and a wide range of sizes that protrude from the dendrite. These spines are the regions where the postsynaptic biochemical machinery responds to the neurotransmitters. Spines are dynamic structures, changing in size, shape, and number during development and aging. While spines and synapses have inspired neuromorphic engineering, the biophysical events underlying synaptic and structural plasticity of single spines remain poorly understood. Our current focus is on understanding the biophysical events underlying structural plasticity. I will discuss recent efforts from my group — first, a systems biology approach to construct a mathematical model of biochemical signaling and actin-mediated transient spine expansion in response to calcium influx caused by NMDA receptor activation and a series of spatial models to study the role of spine geometry and organelle location within the spine for calcium and cyclic AMP signaling. Second, I will discuss how mechanics of membrane-cytoskeleton interactions can give insight into spine shape region. And I will conclude with some new efforts in using reconstructions from electron microscopy to inform computational domains. I will conclude with how geometry and mechanics plays an important role in our understanding of fundamental biological phenomena and some general ideas on bio-inspired engineering.

In this talk, I will describe my lab’s recent efforts to understand the design principles of the active, soft materials that drive cell morphogenesis. In particular, we are interested in how collections of myosin II motors and actin polymers generate, relax, sense and adapt to mechanical force. I will discuss how motor-filament interactions lead to either distributed extensile or contractile stresses as the mechanics of the system changes from fluid to solid. Using optical control of motors, we are now exploring how spatially structured stress can be used to drive local flows and motion. If time, I will also describe how feedbacks between local geometry and activity can be harnessed to drive morphogenetic changes in model systems.