Chemical Pumps and Flexible Sheets Spontaneously Form Self-Regulating Oscillators in Solution

Chemical pumps and flexible sheets spontaneously form self-regulating oscillators in solution

Raj Kumar Mannaa, Oleg E. Shklyaeva, and Anna C. Balazsa,1

aDepartment of Chemical Engineering, University of Pittsburgh, Pittsburgh, PA 15260

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved February 13, 2021 (received for review November 4, 2020) The synchronization of self-oscillating systems is vital to various active sheets encompass a level of autonomous spatiotemporal biological functions, from the coordinated contraction of heart activity that extends the limited repertoire of self-oscillating soft muscle to the self-organization of slime molds. Through modeling, materials and facilitates the fabrication of autonomous, self- we design bioinspired materials systems that spontaneously form regulating soft robots. shape-changing self-oscillators, which communicate to synchro- The mechanism driving the oscillatory, shape-changing be- nize both their temporal and spatial behavior. Here, catalytic re- havior involves a distinctive combination of chemomechanical actions at the bottom of a fluid-filled chamber and on mobile, transduction, the confinement of the host fluid, and a feedback flexible sheets generate the energy to “pump” the surrounding loop. Fig. 1A shows the key components in the system: a fluid- fluid, which also transports the immersed sheets. The sheets exert filled chamber that contains a surface-anchored catalytic patch a force on the fluid that modifies the flow, which in turn affects and a deformable sheet (Fig. 1A). A fixed concentration of re- the shape and movement of the flexible sheets. This feedback actants is added to the solution to initiate the catalytic reaction enables a single coated (active) and even an uncoated (passive) at the central patch. The energy released from this reaction is sheet to undergo self-oscillation, displaying different oscillatory converted into the mechanical motion (flow) of the surrounding modes with increases in the catalytic reaction rate. Two sheets fluid. This constitutes the chemomechanical transduction vital to (active or passive) introduce excluded volume, steric interactions. the observed behavior. Since the system is symmetric about the This distinctive combination of the hydrodynamic, fluid–structure, central patch, the fluid flow occurs about each side of this patch and steric interactions causes the sheets to form coupled oscilla- (Fig. 1A). These symmetric streams eventually hit the confining tors, whose motion is synchronized in time and space. We develop walls, driving the fluid to circulate and form two convective rolls APPLIED PHYSICAL SCIENCES a heuristic model that rationalizes this behavior. These coupled (as detailed in the results section). Hence, the confinement of self-oscillators exhibit rich and tunable phase dynamics, which de- the fluid is another critical component for these self-oscillations. pends on the sheets’ initial placement, coverage by catalyst and The third critical component is the feedback loop that arises relative size. Moreover, through variations in the reactant concen- from the fluid–structure interactions between the circulating tration, the system can switch between the different oscillatory fluid and the compliant sheet (whether the sheet is passive or modes. This breadth of dynamic behavior expands the functional- chemically active). The fluid transports the sheet within the ity of the coupled oscillators, enabling soft robots to display a chamber, but this sheet also exerts a force on the fluid that variety of self-sustained, self-regulating moves. modifies the flow. The modified flow in turn affects the movement of the sheet. This feedback mechanism contributes to the chemically active | reconfigurable sheets | self-oscillating system | oscillatory behavior; the magnitude of the effect depends on the shape-changing coupled oscillators | spatiotemporal synchronization geometry and flexibility of the sheet. The examples below illustrate of coupled oscillators Significance elf-oscillating chemical reactions transduce a constant, non- Speriodic input of energy into sustained periodic motion. Such Using computational modeling, we designed a self-oscillating self-oscillating chemistry is resplendent in biology, enabling the materials system that is driven by a nonperiodic chemical re- firing of neurons, the beating of the heart, and the cyclic be- action to undergo both periodic shape changes and motion. havior of predator–prey relationships (1). The development of Catalytic reactions in a fluid-filled microchamber drive the self-oscillating, shape-changing materials would hasten the de- movement of the fluid and immersed flexible sheets. The fluid velopment of soft robots that autonomously perform self-sustained affects the sheets’ shape, and the sheets affect the fluid flow. work and controllable movement (2, 3). With few exceptions (4–6), This feedback enables remarkably rich and controllable oscil- however, the creation of synthetic self-oscillating materials re- latory behavior: a single sheet fishtails periodically across the mains a significant challenge. Most candidate soft materials only chamber or circulates continuously within a narrow region. exhibited periodic behavior when exposed to variations in the Two sheets form coupled oscillators displaying not only syn- constant energy input (e.g., from changes in illumination, heat, chronized temporal behavior, but also unique, coordinated humidity, or pH) (7–12). The rare exceptions include soft ma- morphological reconfigurations. These oscillators enable de- terials that incorporate one of three intrinsically self-oscillatory velopment of soft robots that operate through an inherent chemical reactions, e.g., self-oscillating gels driven by the Belouzov– coupling of chemistry and motion, permitting novel autono- Zhabotinsky reaction (13–17). Herein, we use computational mod- mous and self-regulating behavior. eling to design chemically driven, flexible micro- to millimeter-sized sheets powered by nonoscillatory chemical reactions that form Author contributions: R.K.M., O.E.S., and A.C.B. designed research, performed research, self-oscillating, shape-changing systems in solution. A single two- analyzed data, and wrote the paper. dimensional sheet spontaneously morphs into a three-dimensional The authors declare no competing interest. structure that moves periodically in time. Two sheets form cou- This article is a PNAS Direct Submission. pled oscillators that communicate to synchronize both their mo- Published under the PNAS license. tion and morphology. While biological and synthetic systems can 1To whom correspondence may be addressed. Email: [email protected]. exhibit self-organized motion and synchronization, there are few This article contains supporting information online at https://www.pnas.org/lookup/suppl/ systems that exhibit coordinated spatial movement, structural doi:10.1073/pnas.2022987118/-/DCSupplemental. change, and temporal synchronization (18). These responsive, Published March 15, 2021.

PNAS 2021 Vol. 118 No. 12 e2022987118 https://doi.org/10.1073/pnas.2022987118 | 1of8 Downloaded by guest on September 27, 2021 Fig. 1. Self-oscillations of a passive sheet. (A) Schematic of the fluidic chamber containing a chemical pump (marked by red rectangular region) and a passive elastic sheet (in blue). (Inset) The network of nodes (marked by blue dots) that form the sheet and the flexible bonds between nodes (white lines). Stretching 2 and bending moduli of the sheet are κs = 60pN and κb = 7.2pNmm , respectively. The sheet is 1.95mm30.6mm30.26mm in size. (B) Low reaction rate: patch −2 −1 rm = 52μmol m · s (Movie S1). Black arrows indicate the direction and magnitude of the fluid flow; the arrows on the left vertical wall reveal the flows within a vertical plane passing through the center of the chamber. Arrows on the right sidewall indicate the flow in the orthogonal vertical plane. Color bar indicates the concentration of H2O2 in the solution. (C) Height (zs) of the center of the sheet. Color bar indicates time. (D and E) High reaction rate: patch −2 −1 rm = 72μmol m · s .(F) Center of the sheet oscillates back and forth across the catalytic pump. (G and H) At the highest reaction rate, patch −2 −1 rm = 96μmol m · s , the sheet simply circulates within the right half of the domain. (I) Corresponding temporal motion of the center of the sheet. Times for the sheet configurations are marked on the bottom of the rightmost column.

the complex dynamics that emerges due to interactions of these for the observed pumping velocities in a number of systems in- three critical components and provide new systems for uncov- volving catalytic patches on an immobile surface and mobile, ering factors that regulate nonequilibrium behavior. spherical particles (chemical motors) (20–22). For example, we predicted and confirmed experimentally that catalytic reactions Computational Model on the bottom wall of a chamber generate solutal buoyance forces, The surface-anchored catalytic patch (Fig. 1A) works as a which deliver microparticle cargo to specified regions in micro- “ ” chemical pump that drives the fluid flows along the bottom chambers and consequently, the cargo is deposited around a surface toward or away from the patch, depending on the nature specific position on the surface (22). We also predicted and ex- of the chemical reaction. This flow occurs due to solutal buoyancy perimentally validated that the coordination among three enzy- (19, 20). Specifically, the enzymes on the catalytic patch decom- matic pumps leads to the formation of circuit containing a bifurcating pose the chemical reactants into products, which can be denser or stream, which simultaneously delivers microparticles to two dif- less dense than the reactants and thus alter the local density of the ferent locations (23). In the case of chemical motors, we showed fluid. The density variation in the solution gives rise to a buoyancy that microparticles uniformly coated with enzymes in solution b = ρ ∑β ρ force per unit volume, F g 0 iCi,where 0 is the solvent could undergo self-sustained motion in the presence of the appro- density, g is gravitational acceleration, Ci is the concentration of priate reactant (24). These studies were also verified by subsequent 1 ∂ρ chemical species i,andβ = ρ ∂ are the corresponding solutal i 0 Ci experimental studies (25). While there have yet to be experiments expansion coefficients. Fb drives the spontaneous fluid motion. on chemically active sheets, we have developed an analytical theory In previous studies, we combined modeling and experiments that showed quantitative agreement with our computational model to show that the solutal buoyancy forces are primarily responsible for the folding of a single sheet in solution (26).

2of8 | PNAS Manna et al. https://doi.org/10.1073/pnas.2022987118 Chemical pumps and flexible sheets spontaneously form self-regulating oscillators in solution Downloaded by guest on September 27, 2021 patch sheet In the case of the sheets, the fluid flow alters not only the patch = rm Ci sheet = rm Ci ’ Kd , Kd . [5] position but also the sheet s shape. The sheet is modeled as a KM + Ci KM + Ci single-layer network of nodes, with positions rk, that are inter- – connected by elastic rods (Fig. 1A, Inset and SI Appendix,Section KM (in units of molarity, M) is the Michaelis Menten constant. patch sheet I). The nodes of active sheets are uniformly coated with a catalyst rm and rm are the maximal reaction rates at the chemical pump and active sheets, respectively, and are the product of the and generate buoyancy forces. The passive sheets are uncoated −1 reaction rate per molecule of enzyme ke (s ) and the areal con- and hence, in this case, the buoyancy forces originate solely from − centration of enzyme, [E](mol m 2). the surface-anchored chemical pump. The sheet nodes experi- We use no-slip boundary conditions for the fluid flow (u = 0) sheet = e + s + g ence body forces, F F F F , which are the respective and no-flux conditions for chemical Ci at the confining walls of elastic forces, steric forces, and gravity. The elastic forces Fe are the chamber. The numerical methods for solving the governing characterized by the stretching (κs) and bending (κb) moduli and equations (Eqs. 1–4) with the specific boundary conditions are are governed by the linear constitutive relations for a Kirchhoff described in SI Appendix. The parameters relevant to chemical rod (27). The steric forces on the sheet, Fs, are the sum of the reactions are also given in SI Appendix, Tables S1 and S2. “ – ” nn node node (nn) repulsion between two nodes, F , and repulsion Results: Chemical Pump and Elastic Sheets as a nw between nodes and any of the six confining walls, F . These steric Self-Oscillatory System forces are calculated as the gradient of Morse potential (SI Appen- Self-Oscillation of a Passive Elastic Sheet. Fig. 1A shows the simplest dix). Gravity acting on the sheet is described by Fg = V(ρ − ρ )g, s 0 system considered here: a centrally located chemical pump coated where V istheeffectivevolumeofeachnode.Thedensityofthe with the enzyme catalase and a passive elastic sheet, which lies ρ sheet ( s) is assumed to be greater than the density of the solvent orthogonal to the pump, with its left end crossing two units into (ρ ). In the absence of reactants, gravity drives the sheet to sediment 0 the width of the patch. Initially, hydrogen peroxide (H2O2)is to the bottom of the chamber. added to the confined aqueous solution. Catalase decomposes The dynamic interactions between the elastic sheet and flow- hydrogen peroxide into the lighter products, water (H2O) and ing fluid are described by the following coupled equations: the oxygen (O2), continuity and Navier–Stokes [in the Boussinesq approximation Catalase 1 (28)] equations for the dynamics of an incompressible flow; the H O ̅̅̅→H O + O . [6] 2 2 2 2 2 equation for the advection, diffusion, and reaction of the dis- APPLIED PHYSICAL SCIENCES solved chemical species C ; and the equation for the velocity of i The solutal expansion coefficient for oxygen is approximately an the nodes of the elastic sheet. The respective equations are order of magnitude smaller (22) than that for hydrogen peroxide, ∇ · u = 0, [1] and hence, we neglect its contribution to the density variation in the solution. The reaction rates are sufficiently low that the formation of O2 bubbles can be ignored. Since the temperature ∂ u 1 1 + ( · ∇) =− ∇ + ν ∇2 + increase due to the above chemical reaction is relatively small ∂ u u ρ p u ρ F, [2] t 0 0 and the thermal expansion coefficients are significantly smaller than the solutal expansion coefficients, we neglect the density variation due to temperature changes and treat the system as = ρ ∑β + 1 ()e + ∑ nw + ∑ nn +()ρ − ρ where F g 0 iCi F F F s 0 g, isothermal (22). V ⏟̅̅̅̅̅̅̅̅̅̅̅̅̅⏞⏞̅̅̅̅̅̅̅̅̅̅̅̅̅⏟ ⏟⏞⏞⏟ Since the products of the reaction are less dense than H2O2, sheet solutalbuoyancy the product-rich fluid rises upward. Due to the continuity of the confined fluid, and the symmetry of the physical setup, the flow ∂ C Ns near the top of the chamber splits into two equal streams. Each i + (u · ∇) C = D ∇2 C ± SpatchK patch ±sKsheet ∑ δ(r − r) , stream flows downward along the closest sidewall and back to- ∂ t i i i d d k ⏟̅̅⏞⏞̅̅⏟ ⏟̅̅̅̅̅̅̅̅̅̅⏞⏞̅̅̅̅̅̅̅̅̅̅⏟k=1 ward the center of the patch, forming “inward” flow at the bot- patch sheet tom surface of the chamber. [3] Fig. 1B shows the steady-state conformation of the passive sheet for patch = μ −2 · −1 the lowest reaction rate considered here (rm 52 mol m s ). ∂ r Driven by the underlying catalysis, the fluid velocity is highest k = u. [4] ∂ t above the center of the catalyst-coated patch. This flow drags the sheet toward to the left, i.e., closer to the patch’s center. At early Here, u and p (in Eq. 2) are the respective local fluid velocity and times, the rising fluid (forming the inward flow) drags the sheet pressure, ν is the kinematic viscosity of the fluid, ∇ is the spatial upward and increases the sheet height (zs) as the sheet crosses gradient operator. The body force (per unit volume) F arises the patch (Fig. 1 C, Top). The restoring force from the elastic from solutal buoyancy and the force exerted by the sheet nodes sheet (i.e., the fluid–structure interaction) and the fluid drag on the fluid (the fluid–structure interactions). The immersed resist this motion. Within a relatively short time, the system at- boundary method (27) is used to treat the latter interactions. tains a steady state, where the competing forces are equilibrated. The ith reagent of concentration Ci diffuses with the diffusion This rapid equilibration suggests that forces arising from the constant Di. elasticity of the sheet (determined by κs and κb) and hence, the The chemical is consumed or produced at the chemical pump material properties, play a significant role in establishing the fi- patch patch patch nal structure (as shown in Fig. 2). with a reaction rate S Kd , where S is the surface of area of the patch. For active sheets, chemical reactions occur at the At the steady state, the sheet forms a stable spanning bridge patch (Fig. 1B) centered about the patch width. The bridge height position of catalytic node rk with a reaction rate given by sKd , attains a maximum value relatively quickly and remains constant where s is the surface area per node. The catalytic reactions are in time (Fig. 1C). Importantly, the flow is necessary for initiating modeled using Michaelis–Menten reaction rates (29) and the the dynamic behavior and raising the height of the bridge. The reaction rates are flow profiles (Fig. 1B) at the steady state are seen to be symmetric

Manna et al. PNAS | 3of8 Chemical pumps and flexible sheets spontaneously form self-regulating oscillators in https://doi.org/10.1073/pnas.2022987118 solution Downloaded by guest on September 27, 2021 behavior is most pronounced. For all tested values of the bending modulus, systematic increases in the reaction rate drive the sheet from the steady-state bridge configuration (S), to oscillating across the patch (O2), and ultimately to oscillating in one half domain (O1). The observed oscillations remain stable for a few hours (until the reactant is consumed). Notably, the frequency of the oscillations is constant for a fixed patch rm . It can, however, be tuned by altering this reaction rate at the patch (e.g., by varying the areal concentration of enzyme) and size patch of the sheet (SI Appendix,Fig.S4). Moreover at fixed rm ,the oscillatory behavior can be altered by varying κb or κs since the deformations induced by the fluid will depend on the stiffness of the sheets (SI Appendix,Fig.S4). In the above examples, we used the enzyme catalase to generate the activity. Similar behavior could be produced with other enzymes, such as acid phosphatase and glucose oxidase. Due to the patterns of the fluid flow created by the latter enzymes (26, 30, 31), they would also prompt the oscillatory motion of the sheet.

Self-Oscillation of a Half-Coated, Active Sheet. The oscillatory behavior of a single sheet can be altered by coating a portion of it with catalase (Fig. 3A and Movie S2). Now the catalytic reaction (Eq. 6) occurs on both the patch and the coated area on the sheet. Hence, the state diagram (Fig. 3B) depends on both re- patch sheet κ κ action rates, rm and rm (at fixed values of b and s). Fig. 3 A and B show the scenario where a half-coated sheet (marked in red) Fig. 2. State diagram of the stationary and oscillatory states of a passive sheet as a function of the bending stiffness of the sheet and catalytic reac- is placed on one side of the catalytic patch. The active half of the patch tion rate. Top panels (from left) show the typical trajectory of the center of sheet also generates flow and thus, the rm necessary to induce the sheet for a stationary bridge state (S), a side-to-side oscillatory state (O ), sheet 2 side-to-side oscillations (O2) decreases with an increase of rm .In and a state in which a sheet oscillates in one half of the domain (O ). The S, 1 contrast to the passive sheet, the oscillations in the O2 state of the O2, and O1 states are represented by circle, triangle, and square, respectively. half-coated, active sheet displays different oscillation amplitudes at The colors provide a guide to the eye. Here, the stretching moduli of the each side of the patch, one corresponding to the active half being sheet is κ = 60pN. s above the patch and another when the passive portion is above the patch. This patterning also affects the trajectory of the side-to-side oscillations; the center of the trajectory is now shifted toward right about the center of the chamber. The letter “S” indicates this of the patch (SI Appendix,Fig.S5), indicating that the flow gener- steady state in the state diagram in Fig. 2. ated through the interactions between the patch and active part of patch = μ −2 · −1 An increase in the reaction rate (rm 72 mol m s ) the sheet pulls the passive sheet to the right. The similar trend is corresponds to an increase in the velocity of the generated flow observed for the transition from the O2 to O1 state. These trends (as evident through Eqs. 2 and 3). Now, the forces originating are relatively insensitive to the initial location of the sheet (SI Ap- from the flow and the elastic sheet are comparable in value. This pendix,Fig.S6). As shown in SI Appendix,Fig.S7, the state diagram is evident from the “tug of war” between these competing forces can be tuned by coating larger portions of the sheet or by changing that causes the sheet to oscillate, moving back and forth across the bending and stretching moduli. the patch (Fig. 1 D and E). In each cycle, forces exerted by the Two Passive Sheets Act as Autonomously Coupled Oscillations. rectangular sheet on the fluid compress a significant portion of The the nearest convective roll (SI Appendix, Fig. S2); in turn, the single-sheet studies provide insight for understanding self-oscillatory momentum transfer from the fluid to the sheet drives the sheet systems involving two or more sheets, which respond not only to the into the adjacent compartment (Movie S1). The trajectory of this motion exhibits periodic behavior with time (Fig. 1F); this state is labeled “O2” in Fig. 2. Notably, a sphere of a comparable size (diameter= 0.6 mm) does not generate fluid–structure interaction forces that span across the patch and generated convective roll (see SI Appendix, Section IV); this compression requires the long arm of the sheet. Consequently, the sphere does not oscillate across the patch, but simply circulates in one half of the chamber. patch = μ −2 · −1 For a further increase of the reaction rate (rm 96 mol m s ), the fluid velocities generated at the pump are sufficiently large that they dominate the system’s behavior. The passive sheet now simply oscillates about the center of a half domain (Fig. 1 G and Fig. 3. Self-oscillation of a partially coated active sheet. (A) Schematic view H, SI Appendix, Fig. S3, and Movie S1). Fig. 1I shows the elliptic of a fluid chamber containing a catalytic pump and an active elastic sheet. trajectory of the center of the sheet as it undergoes this sustained The red region of the sheet is made chemically active by coating it with “ ” catalase. (B) State diagram of the active sheet as a function catalytic reaction periodic motion; this state is labeled O1 in Fig. 2. rate of the patch, rpatch and catalytic reaction rate of the sheet, rsheet. Sym- The state diagram (Fig. 2) is plotted as a function of the m m bols represent the following: stationary state (circle), O2 state (triangle), and bending modulus of the sheet and catalytic reaction rate at the O1 state (square). Here, the bending and stretching moduli of the sheet are 2 patch, and delineates the regions where the different oscillatory κb = 7.2pNmm and κs = 60pN, respectively.

4of8 | PNAS Manna et al. https://doi.org/10.1073/pnas.2022987118 Chemical pumps and flexible sheets spontaneously form self-regulating oscillators in solution Downloaded by guest on September 27, 2021 flow generated by the pump, but also to the presence of addi- but with a finite phase difference (SI Appendix, Fig. S9). We tional sheets. In the simplest case, two passive sheets are placed verified this behavior by perturbing the initial position of the symmetrically on either side of the catalytic patch (Fig. 4A). In sheet with respect to the patch. Any initial placement led to the each half domain, the chemically generated, inward flow drags a sheets’ out-of-phase synchronization with the same phase dif- sheet upward. The two sheets are coupled not only through the ference, indicating that out-of-phase oscillations represent a hydrodynamic interactions in the intervening fluid, but also stable dynamical state. through the mutual excluded volume interactions. For the low patch = μ −2 · −1 Autonomous Coupled Oscillations of Two Active Sheets. To illustrate reaction rate, rm 52 mol m s , the system reaches a dynamic steady state, forming a stationary towerlike structure the complex oscillatory behavior exhibited by sheets coated with (Fig. 4B), which is stable for relatively long times (Fig. 4C). The catalytic patches, we first consider sheets containing both active fluid dynamics primarily acts to initiate the motion of the sheet; (red) and passive (blue) regions. The initial orientations of the the steric interactions play the dominant role. Steric interactions active regions are symmetric (Fig. 5A) and antisymmetric (Fig. 5B) patch = μ −2 · −1 about the chemical pump. The symmetrically placed sheets oscil- are also dominant at rm 72 mol m s ; they prevent the late in-phase; however, the oscillation amplitudes are relatively back and forth motion evident for the single sheet. Rather, each lower when the passive regions come face-to-face and higher when sheet circulates in the respective half domain. the active portions face each other (Fig. 5G and Movie S4). The The latter circulating behavior also occurs at the highest re- higher amplitudes within the cycle (Fig. 5G) are due to the in- patch = μ −2 · −1 action rate, rm 96 mol m s , where hydrodynamic inter- creased fluid flow generated by the simultaneous action among actions dominate the behavior of a single sheet (Fig. 1 G–I). At three active regions, two on the sheets and the one on the pump. this reaction rate, a comparison of the oscillation frequency and In the case of the asymmetric placement, the three active regions amplitude for one sheet versus two reveals that for the two-sheet do not work in concert and thus generate nonuniform fluid flows case, the oscillation frequency is lower and the amplitude is about the pump. These flows impose different fluid drags on the higher than for one sheet (SI Appendix, Fig. S8). This difference sheets, causing the sheets to exhibit asynchronous oscillations is due to a reduction in the flow velocity; the fluid exerts more (Fig. 5H and Movie S4). drag and must perform greater mechanical work on two than on For two fully coated sheets, the generated fluid flows are one single sheet. The flow velocity is also reduced due to the symmetric about the pump (Fig. 5F). Consequently, the sheets fluid–structure instructions from the additional sheet. are pulled over the patch at the same time and thus oscillate in- This difference in the one and two passive sheet cases also phase (Fig. 5I). While appearing qualitatively similar, the am- indicates that the sheets affect their mutual dynamic behavior. plitude and frequency of the in-phase oscillations are higher for APPLIED PHYSICAL SCIENCES Indeed, Fig. 4F shows that the two act as coupled oscillators dis- the two fully coated sheets than the two passive sheets. The playing in-phase synchronization (Movie S3). When two sheets are phase differences between the two oscillating active sheets can placed at different distances from the catalytic patch, then after an be altered by changing the relative size of the sheet (SI Appendix, initial period of synchronization, the sheets oscillate in synchrony Fig. S10). Moreover, a difference in oscillation frequencies can

Fig. 4. Autonomous coupled oscillations of two passive sheets. (A) Two passive sheets (marked in blue) are placed on either side of the chemical pump patch −2 −1 (marked in red) in a fluid-filled chamber. (B) At low reaction rate at the catalytic pump, rm = 52μmol m · s , the sheets form a stable towerlike structure (Movie S3). (C) The height (zs) of the center of the sheets remains constant for a long time. The left and right color bars indicate the time elapsed during the patch −2 −1 motion of left and right sheets, respectively. (D and E) For a higher reaction rate, rm = 96μmol m · s , both sheets oscillate about the center of each half domain. (F) The trajectory of the center of the sheets lies either side of the catalytic path (Top). The heights of the center of the sheets as a function of time show are identical and thus indicate that the sheets are synchronized in-phase. The stretching and bending moduli of the sheets are taken as κs = 60pN and 2 κb = 2.2pNmm , respectively.

Manna et al. PNAS | 5of8 Chemical pumps and flexible sheets spontaneously form self-regulating oscillators in https://doi.org/10.1073/pnas.2022987118 solution Downloaded by guest on September 27, 2021 Fig. 5. Autonomous coupled oscillations of two active sheets. (A–C) Two partially coated (red) sheets are initially placed in symmetric (A) and antisymmetric configuration (B) whereas two fully coated sheets are placed in symmetric locations about the patch in C. The corresponding configurations of the sheets at 66 min after the start of the catalytic reaction are shown in D–F (Movie S4). Two partially coated sheets exhibit in-phase synchronized oscillations for symmetric initial placement (G) and display asynchronous oscillations for asymmetric initial placement (H). Two symmetrically placed, fully coated sheets 2 oscillate in a phase-locked state with zero phase difference (I). The bending and stretching moduli of the sheets are κb = 2.2pNmm and κs = 60pN, re- patch = μ −2 · −1 sheet = μ −2 · −1 spectively. Reaction rates for the patch and sheets are rm 90 mol m s and rm 80 mol m s .

be tuned by changing the rate of reaction on one of the sheets observed herein, this coupling mechanism, realized through (SI Appendix, Fig. S11). complex hydrodynamic fluid–structure interactions, yields a range of remarkable phenomena ranging from the feedback that drives Discussion single-sheet oscillations, to synchronization–desynchronization To rationalize the synchronization of the two-sheet oscillators, in dynamics of coupled oscillating sheets. SI Appendix we use a heuristic phase model to demonstrate that In summary, we devised a model of chemomechanical energy attractive hydrodynamic interactions can synchronize rotations transduction that gives rise to a new class of self-oscillating ma- of rigid spheres. On the other hand, when the aspect ratio of the terials system. Given a nonoscillating input, the system organizes particle is increased (to an oblate spheroid) the relative increase into mobile oscillators that not only show temporal synchroniza- in the drag forces can give rise to repulsive interactions, which tion, but also exhibit coordinated shape changes. The results yield can induce out-of-phase motion and more complex dynamics. guidelines for realizing yet other examples of self-oscillating Theseresultscanbeappliedtothesheets carried by convective flow active sheets. Namely, changes in the shape, the patterning of in the closed domains. In particular, for a range of parameters, each catalysts on the sheet, the size and shape of the chemical pump, sheet rotates in its respective half domain, with a trajectory that and the geometry of the chamber can all influence the system’s resembles the periodic movement of a sphere. Hence, like the dynamics. Clearly, computational models are necessary to probe spheres, the rotating sheets can also exhibit synchronization of their the rich design space. The studies presented here, however, pin- mutual motion. The deformation of the rotating sheets leads to point necessary conditions for achieving the observed spatiotem- changes in the interactions between oscillators and changes in their poral coordination: an input of chemical reactants that initiate the respective trajectories. As revealed for the case of oblate spheroids, catalytic reaction at the pump, the confinement of the host fluid, this asymmetry can lead to asynchronous behavior. and sufficiently flexible materials that both respond to and perturb The phase behavior of the sheets, however, is more complex the surrounding fluid (the feedback loop). than that of the rigid spheres and spheroids. The rich phase These coupled oscillators can serve as “chemical clocks,” behavior is attributable to the sheet’s flexibility. Namely, the which constitute valuable tools in analytical chemistry (32, 33). sheets can dynamically adjust their configuration in response to The oscillators can also be useful for creating autonomously self- the hydrodynamic forces imposed by the fluid and, in turn, exert regulating soft robots (34), where the material itself provides the forces back on the fluid. The coupling ensures the exchange control. For example, as illustrated in Fig. 1, the material (whether between the kinetic energy of the moving fluid and elastic energy an internal component or the entire robot) can dynamically adapt stored in the deformed configurations of the elastic sheets. As its motion to variations in the chemical environment. Alternatively,

6of8 | PNAS Manna et al. https://doi.org/10.1073/pnas.2022987118 Chemical pumps and flexible sheets spontaneously form self-regulating oscillators in solution Downloaded by guest on September 27, 2021 the material’s motion and transition between different move- worth noting that the typical flexural rigidity of lipid bilayers is − − ments can be controlled by varying the reactant concentration ∼10 18–10 19 Nm (41, 42) and hence, these materials also form (SI Appendix, Fig. S12). Additionally, a specified phase differ- suitable candidates for further studies. ence in oscillatory behavior between coupled oscillators, and hence, parts of the soft robot, can be achieved through the rel- Methods ative placement of the sheets (as in Fig. 5 A and B). In these A lattice Boltzmann method (LBM) with a single relaxation-time D3Q19 ways, the soft robot’s performance is directed through inherent scheme is used to solve the continuity and Navier–Stokes equations (Eqs. 1 coupling of chemistry and motion, enabling an enhanced degree and 2) at each time step of the simulation (43). The equation for advection, of autonomous behavior. diffusion, and reaction of the chemicals (Eq. 3) is solved using a finite- To guide the experimental realization of these systems, we difference approach with a forward-time central-space scheme. The bound- provide order-of-magnitude estimates for the dimensionless ary condition for the fluid flow at the confining walls of the chamber satisfies = numbers that characterize the system. The magnitude of the fluid the no-slip boundary condition (u 0). For the concentration of chemical Ci, we use the no-flux boundary condition at the solid walls of the chamber, flow is characterized by the ratio of the solutal to viscous forces, ∂ βΔ 3 Ci = 0, where, n^ is surface normal pointing into the fluid domain. The im- = g CL ∂n expressed by the dimensionless Grashof number, Gr ν2 ,where – β Δ mersed boundary (IB) approach is used to capture the fluid structure inter- is the coefficient of solutal expansion, and C is the charac- actions between the elastic sheet and fluid (27, 44). Each node of the elastic teristic chemical variation across the domain (see SI Appendix for sheet is represented by a sphere with effective hydrodynamics radius a that more details). The typical Grashof number corresponding to the μ =( πη )−1 ∼ × 2 accounts for a fluid drag characterized by the mobility 6 a .Theforces fluid velocity in Fig. 1G is 2 10 . The elastic properties of the exerted by the nodes of the elastic sheet on the fluid, calculated using the IB sheet are characterized by the dimensionless flexural rigidity, method, provide zero fluid velocities at the discretized nodes of the elastic which is a ratio of the restoring elastic force to the solutal sheet. Therefore, the IB approach approximates no-slip conditions for the fluid ’= ~ = B ds ’ velocities at the nodes, as well as no fluid permeation through the nodes buoyancy force, B ρ βΔ 3, where B is the dimensional flexural 0g CL rigidity of the sheet and ds is the distance between two neighboring constituting the sheet. Since the elastic sheet is composed of one layer of the nodes, the effective thickness of the elastic sheet is equal to the diameter 2a of nodes of the sheet. The typical dimensionless flexural rigidity of the a single node. We keep the thickness of the sheet constant and vary the elastic ∼ × −5 sheet considered in this study is 5 10 . moduli to alter the mechanical properties of the sheet. We also provide estimates of the dimensional, physical values The velocity field u = ()ux , uy , uz computed using the LBM is used to ad- for salient features of the system in SI Appendix,SectionV.Notably, vect the chemical concentration (Eq. 3) and to update the position of nodes there exist both synthetic and biological thin films that exhibit a of the elastic sheet (Eq. 4). The updated concentration field is then used to comparable or lower flexural rigidity than our model sheet and determine the buoyancy forces in Eq. 2. The time-step size, Δt, in the simulation APPLIED PHYSICAL SCIENCES therefore, these materials could enable experimental studies. For is 1.67 × 10−3s. The size of the computational domain is 42Δx × 42Δx × 17Δx, example, ultrathin flexible sheets of functionalized nanoparticles where the lattice Boltzmann unit Δx is 100μm. Thus, the physical dimension of −20 (NPs) exhibit a flexural rigidity of 10KbT ∼ 4 × 10 Nm at room the simulation box is 4mm × 4mm × 1.5mm. The hydrodynamic radius of the temperature (35, 36). The NP composition of these sheets pro- node, a is taken as 1.3Δx. In the discretization of the elastic sheet, the distance vides sites for catalyst attachment, or using the particles them- between two nearest-neighboring nodes is set to 1.5Δx. The lateral dimensions selves as the catalyst [e.g., platinum NPs and other nanozymes of the elastic sheet in Figs. 1–5are1.95mm× 0.6mm and 1.5mm × 0.6mm, re- (37)]. Photo-cross-linkable polymer films (38, 39) constitute an- spectively, and 0.26mm in thickness. The dimension of the catalytic patch is other candidate material for corresponding experimental studies; 0.4mm × 2.5mm. the catalysts can be incorporated into these films by postmodification Data Availability. of reactive functional groups or direct inclusion of metal NPs (39). All study data are included in the article and/or supporting Alternatively, oleosin surfactant proteins can be utilized to form information. sheets (40) that are hundreds of micrometers long and wide, but ACKNOWLEDGMENTS. We gratefully acknowledge funding from Depart- nanoscale in thickness, mirroring the conditions in our models. ment of Energy Grant DE-FG02-90ER45438 and the computational facilities Functional residues permit enzymes and other catalysts to be an- at the Center for Research Computing at the University of Pittsburgh. We chored to these sheets (19), providing the desired active coating. It is also thank Dr. Victor Yashin for helpful discussions.

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8of8 | PNAS Manna et al. https://doi.org/10.1073/pnas.2022987118 Chemical pumps and flexible sheets spontaneously form self-regulating oscillators in solution Downloaded by guest on September 27, 2021