Computer modeling reveals modalities to actuate mutable, active matter

Catalytic reactions on flexible sheets generate fluid flows that transform the shape of the sheet, which in turn modifies the flow. These complex interactions make computer models vital for designing and harnessing these feedback loops to create soft active matter that autonomously performs self-sustained mechanical work.

hydrodynamic forces that act on the sheet, and the force exerted by the sheet on the surrounding fluid (fluid-structure interactions).
Due to their flexibility, SofAM can exhibit the mutability of living systems. For instance, one sheet coated with different catalytic patches can be reconfigured into multiple shapes, depending on the reactant added to the solution, and thus, allow one sample to perform a range of distinct functions 3 . One challenge in effectively utilizing SofAM is establishing routes to harness the inherent feedback loops in the system (Fig. 1a). Namely, researchers must co-design the SofAM and the chemically generated flows so that the dynamic crosstalk between these components yields the desired structural and temporal evolution. These essential fluid-structure interactions, however, depend on a large number of parameters, including the elasticity, size, geometry, and the number of immersed sheets; arrangement of different catalytic patches on each sheet; types of catalytic reactions and reaction networks (e.g., cascades of coupled reactions); dimensions of the chamber, and presence of catalytic patches on the chamber walls. Moreover, the systems not only dissipate energy but also encompass interconversion across different energy domains (e.g., chemical, hydrodynamic, mechanical domains, optical and thermal). With this level of complexity, experimental approaches would necessitate expending considerable synthetic effort and performing numerous investigations to probe the multi-dimensional phase space. Given the time and expense of methodically analyzing the influence of all the different variables, the experimental studies could potentially miss key interactions and limit our understanding of SofAM. Conversely, multi-physics computer models allow researchers to perform systematic studies, which are necessary to describe the chemo-hydro-mechanical behavior of a single entity and uncover new forms of self-assembly among multiple entities. Such systematic studies are vital for the development and effective utilization of SofAM.
Multi-physics simulations have only recently been developed for SofAM [4][5][6][7] . While these models capture salient features of mobile, elastic surfaces, they, however, neglect either the host fluid 7,8 ; reactions occurring in the solution 4 , or the specific chemistry involved in the reactions 6 . Building on the latter studies, our group 3 developed a computational approach to solve the coupled equations for (1) flow in a fluid-filled chamber; (2) advection and diffusion of chemicals in the solution and their reactions with deformable materials; (3) dynamic behavior of the nodes that make up the elastic material; and (4) fluid-structure interactions between these nodes and the solution (see Fig. 1a). Notably, such coupled interactions are resplendent in biomaterials. For example, all these steps are present as actin-myosin sheets undergo enzymatic reactions in a solution that cause the contraction of the elastic layer and generation of local fluid flow; the resulting fluid-structure interactions drive the sheets to buckle, as shown in Fig. 1c. In Fig. 1d, metabolic activities in thin films of yeast cells lead to density gradients that propel the surrounding fluid, which breaks the initial film into smaller units and thereby helps the dispersion and propagation of the colony.
Below, we illustrate how computational approaches capturing the above (1)-(4) interconnected processes can facilitate the design of synthetic SofAM that perform chemically driven, mechanical work, and thus serve as chemical gears or selfoscillating chemical clocks. More generally, the results reveal new and contractile forces (bottom panels) generated in the system. Panel a is adapted from ref. 9 , Panel b is adapted from ref. 13 . Panel c is adapted from ref. 14 . Panel d is adapted from ref. 15 . Panel e is adapted from ref. 7 . Panel f is adapted from ref. 6 .
modes of actuating and controlling the transformation of 2D biomimetic soft matter into 3D functional materials.
Actuating active sheets Figure 1a shows a deformable, catalyst-coated sheet that resembles a flat wheel, with spokes that interconnect an outer rim and inner center 9 . The center of the sheet is pinned to restrict its horizontal motion but is free to move in the vertical direction. The entire sheet is coated with the enzyme catalase, with the nodes in black being heavier than the ones in green. The addition of hydrogen peroxide activates the catalyst to decompose H 2 O 2 into the lighter products, water and oxygen, and thereby generate buoyancy-driven fluid flow. The generated flow drives the lighter, central region of the sheet to pop up out of the plane, morphing the 2D layer into a 3D shape. The sheet, in turn, exerts forces on the fluid that modify the flow; in particular, the distended spokes act as blades that "catch" the flow and rotate the 3D object. The computer models allowed us to capture and visualize the latter feedback between the fluid and sheet and design the optimal sheet architecture to maximize this fluid-structure interaction. The computer models thereby reveal new routes for actuating flexible material and achieving functionality that is not possible with chemically active matter that is immobile or mobile and inflexible.
With the addition of cogs to the outer rim, the active structure (in green in Fig. 1a(iii)) operates as gear, which activates the motion of passive (non-coated) units (in black). Namely, the green cogs exert a mechanical force on the neighboring cogwheels and thus rotate the passive units in a controllable direction. These results provide a design rule for creating simple machines that undergo self-sustained motion within fluidic devices.
In addition to chemically generated fluid flows, alternative modeling approaches have revealed other forces that can drive 2D elastic sheets to undergo distinctive locomotion. In particular, chemical waves traveling through a flat sheet can generate nonuniform stresses, which drive the sheet to undergo peristaltic-like motion as it effectively crawls on a dry frictional surface (Fig. 1d). Since the 2D sheet can conform to a geometrically patterned substrate (in contrast to a 1D object), the sheet's movement can be tailored by varying the underlying surface topology.
Beyond the dimensionality of flat sheets, researchers have recently modeled active deformable shells 6,8,10 ; to gain insight into the dynamic behavior of biological membranes 11,12 . Figure 1e shows deformations of an initially spherical, elastic shell composed of active nematic particles. The reorientation of these elongated particles introduces topological defects on the shell's surface and generates the flow of the surrounding fluid. The synergistic interactions among the elastic properties of the shell, the dynamic behavior of the topological defects, and the flowing fluid drive the self-morphing of the shell into distinct nonspherical shapes (Fig. 1e). These results shed light on the morphogenesis of biological cells and tissues and can guide the design of bioinspired soft materials.

Driving the self-organization of self-oscillating SofAM
The synchronization of self-oscillating systems is vital to various biological functions, from the coordinated contraction of heart muscle to the self-organization of slime molds. Through modeling, Balazs and co-authors designed bioinspired materials systems that spontaneously form shape-changing, self-oscillators, which communicate to synchronize both their temporal and spatial behavior 13 . The approach utilizes both immobile and mobile surfaces in the microchamber. The green regions in Fig. 1b show that a portion of the bottom, stationary wall, and two movable sheets, which initially lie on either side of this patch, are coated with catalase. With the addition of H 2 O 2 , the active, immobile patch and the sheets generate an inward fluid flow (marked in black) centered at the patch. The combined hydrodynamic forces pull the sheets toward this center, while the excluded volume interactions keep the sheets apart. Through the inherently coupled interactions, the fluid flow affects the shape and movement of the flexible sheets, which in turn modify the fluid flow. This distinctive feedback, which combines hydrodynamic, steric, and fluid-structure interactions, causes the sheets to form selfregulating coupled oscillators, whose motion is synchronized in time and space. The phase dynamics of the coupled selfoscillators can be tuned by altering the extent of catalyst coverage on either the sheets and/or the surface patch. This breadth of dynamic behavior expands the functionality of the coupled oscillators, enabling soft robots to display a variety of self-sustained, self-regulating moves.

Future challenges
Creating predictive models for controlling the actuation and motion of SofAM will require further developments in theory and simulation. While the above discussion focused on buoyancydriven flow, two or more mechanisms for propelling fluids (such as diffusiophoresis, electrophoresis, and Marangoni effects) can be triggered simultaneously by a single chemical stimulus. Models that capture these simultaneous processes can reveal new means to control the surface and bulk flows independently and thus, provide effective approaches to separately direct the selforganization of the immersed sheets and particles or autonomously sort different sized particles in microchannels 3 .
As described above, the spatiotemporal behavior of SofAM sheets typically involves multiple coordinated, dynamic processes; capturing these processes within computationally feasible timescales introduces new challenges for modeling these systems in 3D. For example, the model of the spontaneous buckling of an acto-myosin sheet (Fig. 1c) encompasses the coordinated contraction of the sheet's edges, the autonomous generation of density gradients driven by these contractions, and the ensuing fluid flow 14 . Modeling this integrated behavior in SofAM to understand and ultimately direct their dynamic assembly could require simulations involving hundreds (or thousands) of actin filaments, myosin motors and cells. More coarse-grained modeling approaches are needed to simulate systems of that size scale in reasonable run times.
The above studies focused on the conversion of chemicals into mechanical energy. Chemical reactions driven by light (or other stimuli) can involve multiple energy domains, and thus require models for photo-chemo-thermo-mechanical transduction. To capture this coupling of energy fields also requires longer simulation run times, as well as accurate models that describe the interaction among all these dynamic processes.
Another challenge for modeling is to simulate cascade reactions that provide a programmable delay between individual chemical reactions. This delay is critical for directing particle motion in multi-step processes. For example, with regulation, a reaction yielding dense products can lead to outward motion (away from a catalytic patch), before the next reaction, which produces less dense products, would lead to inward motion. In other words, coordination between the chemical reactions will modulate directed transport (whereas simultaneous activation of both reactions would cancel out the fluid motion). Simulating cascades involving a number of reactions may require the development of new coarse-grained models, but meeting this goal would allow researchers to design multi-stage processes that perform autonomously in microchambers.