Active matter encompasses different nonequilibrium systems in which individual constituents convert energy into non-conservative forces or motion at the microscale. This review provides an elementary introduction to the role of topology in active matter through experimentally relevant examples. Here, the focus lies on topological defects and topologically protected edge modes with an emphasis on the distinctive properties they acquire in active media. These paradigmatic examples represent two physically distinct classes of phenomena whose robustness can be traced to a common mathematical origin: the presence of topological invariants. These invariants are typically integer numbers that cannot be changed by continuous deformations of the relevant order parameters or physical parameters of the underlying medium. We first explain the mechanisms whereby topological defects self propel and proliferate in active nematics, leading to collective states which can be manipulated by geometry and patterning. Possible implications for active microfluidics and biological tissues are presented. We then illustrate how the propagation of waves in active fluids and solids is affected by the presence of topological invariants characterizing their dispersion relations. We discuss the relevance of these ideas for the design of robotic metamaterials and the properties of active granular and colloidal systems. Open theoretical and experimental challenges are presented as future research prospects.

pdf/topological-active-matter-3pbg45jcmv.pdf

Topological active matter

Conspectus: The ability to navigate in chemical gradients, called chemotaxis, is crucial for the survival of microorganisms. It allows them to find food and to escape from toxins. Many microorganisms can produce the chemicals to which they respond themselves and use chemotaxis for signalling which can be seen as a basic form of communication. Remarkably, the past decade has let to the development of synthetic microswimmers like e.g. autophoretic Janus colloids, which can self-propel through a solvent, analogously to bacteria and other microorganims. The mechanism underlying their self-propulsion involves the production of certain chemicals. The same chemicals involved in the self-propulsion mechanism also act on other microswimmers and bias their swimming direction towards (or away from) the producing microswimmer. Synthetic microswimmers therefore provide a synthetic analogue to chemotactic motile microorganisms. When these interactions are attractive, they commonly lead to clusters, even at low particle density. These clusters may either proceed towards macrophase separation, resembling Dictyostelium aggregation, or, as shown very recently, lead to dynamic clusters of self-limited size (dynamic clustering) as seen in experiments in autophoretic Janus colloids. Besides the classical case where chemical interactions are attractive, this Account discusses, as its main focus, repulsive chemical interactions, which can create a new and less known avenue to pattern formation in active systems leading to a variety of pattern, including clusters which are surrounded by shells of chemicals, travelling waves and more complex continously reshaping patterns. In all these cases `synthetic signalling' can crucially determine the collective behavior of synthetic microswimmer ensembles and can be used as a design principle to create patterns in motile active particles.

Synthetic Chemotaxis and Collective Behavior in Active Matter

Living systems at all scales aggregate in large numbers for a variety of functions including mating, predation, and survival The majority of such systems consist of unconnected individuals that collectively flock, school, or swarm However, some aggregations involve physically entangled individuals, which can confer emergent mechanofunctional material properties to the collective Here, we study in laboratory experiments and rationalize in theoretical and robophysical models the dynamics of physically entangled and motile self-assemblies of 1-cm-long California blackworms (Lumbriculus variegatus, Annelida: Clitellata: Lumbriculidae) Thousands of individual worms form braids with their long, slender, and flexible bodies to make a three-dimensional, soft, and shape-shifting "blob" The blob behaves as a living material capable of mitigating damage and assault from environmental stresses through dynamic shape transformations, including minimizing surface area for survival against desiccation and enabling transport (negative thermotaxis) from hazardous environments (like heat) We specifically focus on the locomotion of the blob to understand how an amorphous entangled ball of worms can break symmetry to move across a substrate We hypothesize that the collective blob displays rudimentary differentiation of function across itself, which when combined with entanglement dynamics facilitates directed persistent blob locomotion To test this, we develop a robophysical model of the worm blobs, which displays emergent locomotion in the collective without sophisticated control or programming of any individual robot The emergent dynamics of the living functional blob and robophysical model can inform the design of additional classes of adaptive mechanofunctional living materials and emergent robotics

Collective dynamics in entangled worm and robot blobs

Active matter encompasses different nonequilibrium systems in which individual constituents convert energy into non-conservative forces or motion at the microscale. This review provides an elementary introduction to the role of topology in active matter through experimentally relevant examples. Here, the focus lies on topological defects and topologically protected edge modes with an emphasis on the distinctive properties they acquire in active media. These paradigmatic examples represent two physically distinct classes of phenomena whose robustness can be traced to a common mathematical origin: the presence of topological invariants. These invariants are typically integer numbers that cannot be changed by continuous deformations of the relevant order parameters or physical parameters of the underlying medium. We first explain the mechanisms whereby topological defects self propel and proliferate in active nematics, leading to collective states which can be manipulated by geometry and patterning. Possible implications for active microfluidics and biological tissues are presented. We then illustrate how the propagation of waves in active fluids and solids is affected by the presence of topological invariants characterizing their dispersion relations. We discuss the relevance of these ideas for the design of robotic metamaterials and the properties of active granular and colloidal systems. Open theoretical and experimental challenges are presented as future research prospects. 

Significance The morphogenesis of two-dimensional bacterial colonies has been well studied. However, little is known about the colony morphologies of bacteria growing in three dimensions, despite the prevalence of three-dimensional environments (e.g., soil, inside hosts) as natural bacterial habitats. Using experiments on bacteria in granular hydrogel matrices, we find that dense multicellular colonies growing in three dimensions undergo a common morphological instability and roughen, adopting a characteristic broccoli-like morphology when they exceed a critical size. Analysis of a continuum “active fluid” model of the expanding colony reveals that this behavior originates from an interplay of competition for nutrients with growth-driven colony expansion, both of which vary spatially. These results shed light on the fundamental biophysical principles underlying growth in three dimensions.

Morphological instability and roughening of growing 3D bacterial colonies

During colony growth, complex interactions regulate the bacterial orientation, leading to the formation of large-scale ordered structures, including topological defects and microdomains. These structures may benefit bacterial strains, providing invasive advantages during colonization. Active matter dynamics of growing colonies drives the emergence of these ordered structures. However, additional biomechanical factors also play a significant role during this process. Here we show that the velocity profile of growing colonies creates strong radial orientation during inward growth when crowded populations invade a closed area. During this process, growth geometry sets virtual confinement and dictates the velocity profile. Herein, flow-induced alignment and torque balance on the rod-shaped bacteria significantly differed, resulting in new stable orientational equilibrium in the radial direction. Our analysis revealed that the dynamics of these orientational defects depend on bacterial length and can promote the survival of the longest bacteria around localized nutritional hot spots. The present results indicate a new mechanism underlying structural order and provide mechanistic insights into the dynamics of bacterial growth on complex surfaces.

Large-scale orientational order in bacterial colonies during inward growth

During colony growth, complex biomechanical interactions regulate the bacterial orientation, leading to the formation of large-scale ordered structures including topological defects. These structures may benefit bacterial strains, providing invasive advantages during colonization. Thus far, the emergence of 1/2 topological defects has been observed and extensively assessed. Here, we experimentally and numerically investigate aster formation in bacterial colonies during inward growth when crowded populations invade a closed area. Herein, the velocity field and torque balance on the rod-shaped bacteria significantly differed, resulting in new stable orientational equilibrium in the radial direction and to the formation of +1 topological defects. The dynamics of these defects depend on bacterial length and can promote the survival of the longest bacteria around localized nutritional hot spots. The present results indicate a new mechanism underlying defect formation and provide mechanistic insights into the dynamics of bacterial growth on complex surfaces.

Formation of self-assembled asters in growing bacterial colonies

Author response: Dynamics of pattern formation and emergence of swarming in Caenorhabditis elegans

Bacterial biofilms collectively develop distinct and ordered structures, including fibers, bundles, and branches. Often, it is unclear how these structural motifs convey specific advantages to bacterial strains under challenging conditions. In oxygen-limited environments, dense bacterial aggregates generally deplete oxygen, which leads to arrest of bacterial growth. However, we observed that biofilm-forming Bacillus subtilis could use branching patterns to escape from these crowded regions by tracking the oxygen gradient. The process depends on the chain-forming ability of the biofilm. As a result of collective branching triggered by bending and mechanical buckling, bacteria can extend from the oxygen-depleted biofilm core to the oxygen available periphery. Remarkably, these bacterial branches are strongly chiral and curve in a clockwise direction. Our analysis revealed that the surface friction and axial rotation of the twisting cell wall break left-right symmetry on solid surfaces and drive chiral bending of bacterial chains. We further observed that the chirality of individual branches could propagate across a large scale and shape the macroscopic morphology of the colony under a limited spatiotemporal growth profile. Taken together, our results provide new insights into how simple physical interactions lead to bacterial collaboration and promote the survival of biofilm-forming strains in challenging environments. Although many biological molecules are structurally chiral, we do not know how this chirality shapes macroscopic morphology. Here we studied the direct relation between the emergence of chiral morphology of the growing bacterial biofilms originating from branching patterns. We found that surface friction and growing cell wall initiates the asymmetric growth dynamics in bacterial biofilms and triggers the large-scale chirality. Interestingly these chiral branches help bacterial biofilms to spread more effectively in oxygen-limited environments. Our results provide new mechanistic insights into how physical interactions control the entire biofilm morphology and bacterial competition. are generally straight (c). Sandwiched configuration can balance the rotational friction that occurs between the axially rotating cell wall and the agar surface. (f) Schematic representation of B. subtilis cell wall twisting in the right-handed direction during elongation. The twisting cell wall rotates the cell body in the axial direction and experiences rotational friction on a flat solid surface. All together right-handed intrinsic chirality of the bacterial cell wall leads to clockwise bending of the growing bacterial chains (seen from the top). ( g) Experimental measurement of bending curvature of the branches shows a constant profile. (h) FEM simulation of the growing rod under constant tip force and surface friction. Growing rods provide similar constant bending curvature. (i) Accumulated stress in a growing elastic rod triggers localized buckling and intrinsic twisting dynamics further leads to the formation of twisting macro fibers. These fibers have a macroscopic twisting periodicity of about Λ = 7 μm and as a result of chain breakage (highlighted in a red circle), they can generate new growing branches.

Bacterial biofilms use chiral branches to escape crowded environments by tracking oxygen gradient

Many electrochemical devices are based on the fundamental process of ion migration and accumulation on surfaces. Complex interplay of molecular properties of ions and device dimensions control the entire process and define the overall dynamics of the system. Particularly, for ionic liquid-based electrolytes it is often not clear which property, and to what extent, contributes to the overall performance of the device. Herein we use X-ray photoelectron spectroscopy (XPS) while the device is under electrical bias. Such a procedure reveals localized electrical potential developments, through binding energy shifts of the atomic core levels, in a chemically specific fashion. Combining it with square-wave AC modulation, the information can also be extended to time domain, and we investigate devices configured as a coplanar capacitor, with an ionic liquid as the electrolyte, in macro-dimensions. Our analysis reveals that a nonlinear voltage profile across the device emerges from spatially non-uniform electrical double layer formation on electrode surfaces. Interestingly the coplanar capacitor has an extremely slow time response which is particularly controlled by IL film thickness. XPS measurements can capture the ion dynamics in the tens of seconds to microseconds range, and reveal that ionic motion is all over the device, including on metallic electrode regions. This behavior can only be attributed to motion in more than one dimension. The ion dynamics can also be faithfully simulated by using a modified PNP equation, taking into account steric effects, and device dimensions. XPS measurements on two devices with different dimensions corroborated and validated the simulation results. The present results propose a new experimental approach and provide new insights into the dynamics of ions across electrochemical devices.

Localized X-ray photoelectron impedance spectroscopy (LoXPIS) for capturing charge dynamics of an ionic liquid electrolyte within an energy storage device.

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