Chemically controlled pattern formation in self-oscillating elastic shells

Significance

Mechanochemical processes drive various biological functions including morphogenesis and morphological changes in responsive synthetic hydrogels. By allowing reactive components to function in an elastic closed shell, autonomous chemical reactions generate microcompartments capable of expanding or contracting or induce buckled patterns. This capability can drive specific functions such as enhancing the rate of diffusion of compartmentalized chemicals or releasing cargos at specific rates. Here, we demonstrate the coupling of mechanical response to autonomous chemical reactions in elastic shells which swell and deswell to generate specific dynamic patterns and morphologies. The reverse feedback from the mechanical input triggers the chemical reaction. Our work inspires the design of responsive shells to enhance the functionality of microcompartments and nanoreactors.

Abstract

Patterns and morphology develop in living systems such as embryos in response to chemical signals. To understand and exploit the interplay of chemical reactions with mechanical transformations, chemomechanical polymer systems have been synthesized by attaching chemicals into hydrogels. In this work, we design autonomous responsive elastic shells that undergo morphological changes induced by chemical reactions. We couple the local mechanical response of the gel with the chemical processes on the shell. This causes swelling and deswelling of the gel, generating diverse morphological changes, including periodic oscillations. We further introduce a mechanical instability and observe buckling–unbuckling dynamics with a response time delay. Moreover, we investigate the mechanical feedback on the chemical reaction and demonstrate the dynamic patterns triggered by an initial deformation. We show the chemical characteristics that account for the shell morphology and discuss the future designs for autonomous responsive materials.

Chemical reactions are key components in morphogenesis and are essential to assemble mechanoresponsive materials with self-healing and adaptive properties (1–4). In particular, the coupling of chemical reactions with responsive gels has offered the possibility to create autonomous materials that convert chemical energy into mechanical energy (5–9) that resemble heart-beating dynamics. The design and synthesis of materials with biological functions is a challenge that involves a delicate balance between structural shape and physiological function. During embryonic development, for example, flat sheets of embryonic cells morph through a series of folds into intricate three-dimensional structures such as branches, tubes, and furrows that build blocks for organs that enable their proper function. Such shape-forming processes are driven by controlling chemical and mechanical signaling events.

Man-made chemo-morph systems, however, are less common than the biological counterpart. Only recently, reconfigurable soft robots have been proposed (10–12). One of the main advantages of soft materials is that they do not need mechanical parts; the motion is driven by external chemophysical signals. Rhythmic swelling and contractions in response to chemical changes have been observed (13–17). The synergy between chemical interactions and locally directed forces leads to selection rules for sizes and shapes that autonomously generate motion (18).

In chemoresponsive soft materials the complex nature of microscopic interaction contributes to a macroscopic morphology (19, 20). The diffusion of reactants, for example, can in principle elicit different responses due to mechanical deformation. As in morphogenetic processes, the flow of information is complex and relies on the interplay among all components. That is, chemically generated stresses induce material deformation that influences the chemical regulators in a closed feedback loop (21–23). Here, we introduce a model to study the shape dynamics triggered by autonomous chemical reactions in closed elastic shells. We focus on chemical changes that introduce periodical hydration changes that prompt gel swelling.

We study autonomous gel membrane dynamics triggered by periodic reduction–oxidation reactions that generate membrane hydration heterogeneities. Instead of considering two-dimensional elastic models for mechanically heterogeneous crystalline small shells that buckled into polyhedral (24), we consider thin membrane elasticity models that generate wrinkling by imposing the metric (25, 26). That is, in order to include the response to the chemical field on the membrane we integrate the chemical field on a non-Euclidean shell elasticity model (27, 28) and in such way investigate the local hydration impact on membrane morphological changes. We simultaneously incorporate the chemical dilution due to stretching and dynamics associated with the deformed gel and study the mechanochemical feedback loop. We report a wide variety of morphologies and oscillation behavior that indicates a robust mechanism for the future design of chemoresponsive gels with desired functionality.

Chemical Activated Surfaces

We consider a chemoresponsive polymeric thin elastic shell of radius $R$ and thickness $h\ll L$ with linear elastic response (29). The shell is represented by a two-dimensional midsurface smooth function with no overhangs. The elastic energy associated with the deformation of the midsurface of the shell can be expressed as (27, 30)

$$\begin{array}{ll}\hfill E_{elastic}\left(g_{\alpha \beta },{\overline{g}}_{\alpha \beta }\right)& =\frac{1}{2}\int dĀ{\mathcal{A}}^{ijkl}{u}_{ij}{u}_{kl}\hfill \\ \hfill & \hspace{1em}+\frac{1}{2}\int dĀ\left(\kappa {\left(H-2H_0\right)}^2+2{\kappa }_{g}G\right).\hfill \end{array}$$ [1]

The first term in Eq. 1 represents the stretching energy, where $u_{\alpha \beta }=\frac{1}{2}\left(g_{\alpha \beta }-{\overline{g}}_{\alpha \beta }\right)$ is the strain tensor, $g_{\alpha \beta }$ (${\overline{g}}_{\alpha \beta }$) is the actual (target) metric tensor, $dĀ=\sqrt{\text{det}\overline{g}}dxdy$ is the target area element, ${\mathcal{A}}^{\alpha \beta \gamma \delta }$ is the elastic tensor, and summation over pairs of repeated indices is assumed. For an isotropic material, ${\mathcal{A}}^{\alpha \beta \gamma \delta }==\frac{Eh}{1-{\nu }^2}\left(\nu {\overline{g}}^{ij}{\overline{g}}^{kl}+\left(1-\nu \right){\overline{g}}^{ik}{\overline{g}}^{jl}\right)$, where $E$ is the Young’s modulus and $\nu$ is the Poisson ratio and ${\overline{g}}^{\alpha \gamma }{\overline{g}}_{\gamma \beta }={\delta }_{\beta }^{\alpha }$. Moreover, the second term accounts for bending energy with the mean curvature $H$, spontaneous curvature $H_0$, and the Gaussian curvature $G$ associated with $\kappa =E{h}^3/12\left(1-{\nu }^2\right)$ and ${\kappa }_g=\left(\nu -1\right)\kappa$ denote the bending and Gaussian rigidity, respectively.

Autonomous Oscillatory Chemical Reactions.

Pattern formation is widespread in nature and our daily life, from the formation of animal skin patterns to the design of material and robotics (31–34). In particular, chemical pattern formation is an important process in biology since it ensures the correct placement and development of tissues within a body (35). In general, pattern-forming systems can be classified into nonautonomous and autonomous (self-organized) patterns. Nonautonomous systems respond to environmental clues and signaling such as chemical gradients. Autonomous chemical patterns, on the other hand, are formed by a complex reaction–diffusion system that involves an activator (autocatalytic chemical) and an inhibitor that slows down the production of the chemicals, which are coupled via a diffusion mechanism for both species (36).

One of the most paradigmatic cases of oscillating pattern formation in gels is the Belousov–Zhabotinsky (BZ) reaction (5, 14, 37). The BZ reaction is a catalytic oxidation reaction of organic acid by bromate anion. The particular choice of the acid, the organic species, or the catalyst give rise to diverse variations of the BZ reaction that involve numerous elementary steps. In the simplest form, the diffusion–reaction system consists of five coupled reactions with both autocatalytic and negative feedback loop steps that can be accurately modeled by the time–space Oregonator model (38, 39) (see SI Appendix, Eqs. S1–S3),

$$\begin{array}{ll}\hfill \frac{du}{dt}& =ϵD_u{\mathrm{\Delta }}_{M}u+u-u^2+fv\frac{q-u}{q+u}\hfill \\ \hfill \frac{dv}{dt}& =ϵ\left(u-v\right),\hfill \end{array}$$ [2]

where $u$ is the activator, denoting the concentration of the reactant, and $v$ is the oxidized form of the catalyst; the dimensionless variables $ϵ$, $q$ are defined through the rate constants and the reservoir concentration. In addition, stoichiometric factor $f$ depends on the organic chemistry involved. Moreover, to ensure the existence of oscillations its value has to be in a certain range. Finally, $D_u$ is the bromous acid diffusion coefficient (see SI Appendix, Table S2). Note that we have taken into account that oxidized catalyst $v$ is moving together with the polymer gel (40).

Mechanochemical Shell Dynamics.

Following ref. 26 we discretize Eq. 1 by triangulating the shell surface. We assume overdamped Langevin dynamics and solve the set of first-order equations for the position of each vertex ${\mathbf{r}}_i$ and discrete metric of each triangle $g_{\alpha \beta }$:

$$\gamma {\dot{r}}_i=-{\nabla }_{{\mathbf{r}}_i}{E}_{elastic}\left(g_{\alpha \beta },{\overline{g}}_{\alpha \beta }\right)+{\mathit{\eta }}_i\left(t\right),$$ [3]

where ${\mathit{\eta }}_i\left(t\right)\in {\mathbb{R}}^3$ is a weak random noise, obeying $⟨{\mathit{\eta }}_i⟩=0$ and $⟨{\eta }_i^m\left(t\right){\eta }_j^n\left(t^{\prime }\right)⟩=\sqrt{2\gamma k_{B}T}{\delta }_{ij}{\delta }_{mn}\delta \left(t-t^{\prime }\right)$ with $m,n\in \left\{x,y,z\right\}$. Here, $T$ is the temperature and $\gamma$ is the friction coefficient, which we assume to be constant. All of the mechanochemical coupling is modeled through the target metric ${\overline{g}}_{\alpha \beta }$, which depends on the chemicals ($v$) embedded in the shell. Eq. 3 is integrated numerically using a standard first-order Euler–Maruyama discretization scheme keeping connectivity of the triangulation fixed. Finally, Eq. 2 is integrated numerically using discrete exterior calculus operators in space for triangular meshes and explicit discretization in time (see Materials and Methods and Eq. 7).

Dynamic Shell Morphology by Active-Chemical Growth

Active Structural Remodeling.

To understand how shell morphology is affected by chemical-induced stresses we first investigate mechanical pattern formation given by a conformal active remodeling step,

$${\overline{g}}_{\alpha \beta }=M{\overline{g}}_{\alpha \beta ,t=0},$$ [4]

where $M$ is the growth ratio (Fig. 1); in this way, the strain field is generated on the shell and it buckles when the shell volume is kept constant. For small $M$, the shells have spherical shapes with no apparent morphological changes. As the target area increases, however, the shells are under extensive compression where the elastic energy is accumulated until the symmetry is broken and the residual stresses are released by the formation of folds and wrinkles (26, 41). The undulation wavelength is proportional to the thickness, which scales as the ratio of bending rigidity to stretching rigidity.

Fig. 1.

An illustration of morphologies a spherical shell of thickness $h$ and radius $R$ as a function of the growth ratio, $M$ (Eq. 4). The morphologies are minimum energy (Eq. 1) configurations that are calculated by gradually increasing $M$ while keeping the total volume fixed. For small $M$, the shells maintain their spherical shape; however, as the growth ratio $M$ increases, the accumulated elastic energy of the shell (Inset) is released and the shell buckles to form wrinkles on its surface. The color scheme of the plots represents the local mean curvature with negative value on the concave faces and positive on the polyhedron edges. (Inset) Total elastic energy, $E_{\text{elastic}}$, as a function of $M$ for various shell thicknesses. The sharp kink in the plot represents the buckling point.

Oscillatory Mechanochemical Coupling.

The previous example provides a clear path of how to control morphological changes by induced strain fields. In chemoresponsive polymeric shells such changes are autonomously controlled by the covalent nature of the catalyst $v$ bond to the polymer gel (5, 9, 14). As the chemical process changes $v$, the free energy of the hydrogel changes instantaneously, thereby creating an instantaneous stress on the material. This chemically induced stress is then gradually released by the surrounding solvent as the solvent molecules are sucked into/out of the material. This leads to changes in the volume of the active gel. Upon solving the model for an unconstrained hydrogel disk we find that for small variations of the catalyst $v$ the changes in the target metric can be expressed as an affine transformation where the growth ratio $M\left(t\right)=1+\alpha v\left(t\right)$ with $\alpha$ a material constant that accounts for the intrinsic properties of the hydrogel such as its cross-link density and the interaction energy of the catalyst $v$ with the solvent (SI Appendix, section S4). This comparison with the results of a hydrogel modeled as in references 5–7 reveals that the linear approximation for the growth ratio $M$ holds for gels with Young’s modulus E $\gtrsim$ 10 kPa (SI Appendix, Fig. S1). For the purpose of simulations, this exercise allows us to remodel the physics of chemically active polymer gels as an affine transformation. Therefore, this method allows us to study complex systems with relative ease.

Similarly to actomyosin contraction in cardiac muscles driven by calcium oscillations (42, 43), BZ reactions can induce swelling–deswelling beating by periodic oxidation and reduction of catalyst. The average $u$ and $v$ concentration profiles as a function of time are plotted in Fig. 2A, where the concentration of reactant $u$ and catalyst $v$ periodically oscillate around their stable value $u^{*}$ and $v^{*}$. By coupling the target metric with $v$, we find in the oxidized state the shell hydrates due to an increase in the catalyst net charge (8). Conversely, when the catalyst $v$ is reduced, the shell starts to shrink, inducing swelling–deswelling beating waves (Fig. 2B). Note that the shell is subject to a volume penalty (SI Appendix, section S3C and Table S2) to account for the low solvent permeability of the shell.

Fig. 2.

(A) The plot of the average chemical concentration of reactants $u$ and $v$ on the shell as a function of time. The shell becomes more hydrophilic with an increasing value of $v$, which leads to swelling of the material. Similarly, decreasing the value of $v$ leads to deswelling. (B) Plot of the shell area ($A/A_0$) as a function of time. The plot depicts the direct correspondence between the area ratios and the concentration $v$ (Movie S1).

Interestingly, recent experiments have found swell/deswell and buckling/unbuckling oscillations in synthesized polymersomes, which are cross-linked polymeric shells composed of the catalyst segment and the hydrophilic segment (14). It is clear that stresses generated by chemical remodeling in a uniform shell are not enough to explain autonomous buckling/unbuckling oscillations shown in the chemoresponsive experiments (14). Here, we propose that heterogeneities in the shell thickness can justify such behavior. To prove our hypothesis, we spatially varied the shell thickness (Fig. 3A) to account for shell design imperfections in real experiments. In agreement with polymersome experiments (14), we observe buckling–unbuckling oscillations consistent with changes in the catalyst concentration (Fig. 3B). The shape oscillations can be understood as follows. As the catalyst concentration increases from its minimal values ($t=187\to 200$; Fig. 3C) a strain field generated by changes in the degree of hydration and the shell starts to build up residual stresses (Fig. 3D). For values of the catalyst larger than $v=0.16$ the degree of hydration is such that the shell swells, releasing the residual stresses built and consequently buckles (Fig. 3D, red). In addition to the morphology with a single buckling point (Fig. 3B), we also find multiple buckling seeds as we increase the thinnest thickness of the shell. In this case, two concave buckling sites are observed in the first cycle, which slowly converge to one buckling point (SI Appendix, Fig. S2). The result is consistent with the recent colloidosome experiment, where multiple buckling and moving buckling points are observed analogous to biological cells (44). Compared to the equilibrium case, we find the moving buckling points are likely introduced in the deformation pathway as a kinetic process, where the relaxation time is not sufficient to release the internal stress.

Fig. 3.

Heterogeneous shell dynamics with fast diffusive chemical reaction. (A) Schematic polymeric shell with nonuniform thickness $h$. The spherical shell buckles inward as the polymers become more hydrophilic. (B) A comparison of buckling observed experimentally (Left image reprinted with permission from ref. 14) with simulation results (Right). The colors of the shell denote the value of catalyst concentration, where yellow indicates oxidized state and orange represents reduced state (14). (C) Plot of the catalyst concentration ($v$) and shell asphericity ($⟨\mathrm{\Delta }{R}^2⟩/{⟨R⟩}^2$) as a function of time. The plot demonstrates the time delay between the increase in $v$ and the mechanical buckling. (D) A representation of stress distribution as a function of the polar angle. The shell is under compression with negative stress and tensile with positive stress. The gray lines denote the stresses from time $t=200$ to $t=260$ with increment $dt=2$, and the colored lines are representatives of the second cycle undergoing the buckling transition (Movie S2).

Geometrical Feedback.

In addition, shell deformations give rise to fluctuations in the catalyst concentration (Fig. 4), which can be understood by two main contributions: changes in the local area and the curvature. Changes in the local area modify the chemical concentration (note that concentration c is defined by $c=N/Area$, where $N$ is the number of molecules; Eq. S5) as illustrated in Fig. 4A. As the local area grows, the chemical concentration ($v$) is diluted below the steady value ($v^{*}$) and the chemical reaction is triggered hereon. To test this geometrical feedback, we introduce a small random force on the shell and investigate how area changes give rise to chemical patterns (Fig. 4B). We observe that even the smallest amount of change in the area can trigger significant deviation from $v^{*}$, provoking the onset of the chemical reactions. Since our approach is based on approximations for thin sheets, the chemical dilution due to the changes of thickness is not considered here. However, other curvature effects (such as the mean curvature) are considered within the Laplace operator in curved spaces (45). We note, however, the feedback from the curvature is small compared to the contribution from area changes.

Fig. 4.

Geometrical feedback from local changes in the area. (A) If the number of molecules remains fixed $N$ but the area increases, then the chemical concentration gets diluted, given that $c=N/Area$. Deviation from the steady state of the chemicals can trigger the mechanism of reaction in the Oreganator model. (B) Even random deformations can give rise to oscillatory patterns for a close shell due to local area changes (Movie S3).

Material Design through Mechanochemical Coupling.

Now that we have established the foundation for mechanochemical coupling in polymer shells we focus on possible material design (46). With that intent, in this section we explore the possibility of multiresponsive materials as a mechanism for generating functional morphological shapes. We consider an in silico material with chemical thickness response of the form $h\left(r,t\right)=h_0+\zeta \hspace{0.17em}v\left(r,t\right)$, where $h_0$ and $\zeta$ are chosen so the minimum thickness of the polymer shell is $h_{min}=0.1a$ and the maximum $h_{max}=1.0a$, with $a$ the length unit.

First, we investigate the possibility of generating self-adapting materials by exploring how the shell can mobilize the reactants. We find that the best way to do it is by mass-transfer effects, that is, by changing the ratio of bromide ion to the catalyst ions during the malonic acid oxidation step, or by changing how fast the bromous acid is replaced via the diffusion process, in other words, by changing the stoichiometric factor $f$ or the diffusion coefficient $D_u$ in the reduced system of equations given in Eq. 2.

The computational experiment starts by establishing a stable chemical concentration of $v^{*}$. Then we introduce a local shell deformation by pipette-pressing the shell, $t=0$ in Fig. 5. The local mechanical changes of the shell generate a chemical concentration gradient in the north and south poles of the shell due to a dilution introduced by the expansion of the material.

Fig. 5.

(A–E) The shell morphology dynamics with different diffusion coefficients and stoichiometric factor $f$. The colors represent the catalyst concentration: blue when the polymer is oxidized and orange in a reduced state. The shell is first relaxed with respect to a stable value of catalyst concentration $v*$, followed by a mechanical press. The initial deformation of the shell causes the chemical dilution and generates the reduction wave from the top and the oxidization wave from the bottom. The overall wave directions are opposite for $f=0.6$ and $f=1.0$, leading to a mushroom shape or exocytosis dynamics. The patterns in the boxes periodically appear with schematic illustrations of the thickness shown on the right (Movies S4–S8). (F) The schematic figures to illustrate the thickness distribution of the shells marked in the green boxes.

As the reaction evolves, two primary waves are generated, a reduction wave from the north pole and an oxidization wave from the south pole. The two waves travel oppositely and meet at a counterline (Fig. 5A). For $f=0.6$, the reduction wave out-speeds the oxidation wave ($t=100$), and the combined wave moves downward until the whole shell is under a reduced state ($t=180$). Immediately after, the shell rapidly develops an oxidization wavefront generating local thickness gradients, inducing a mushroom-like morphology with negative Gaussian curvature ($t=220$). As the wavefront approaches the south pole the shell is compressed, accumulating a large concentration of chemicals that generates a quick wave response that moves the deformation to the north pole ($t=370$). As $f$ increases (Fig. 5B) the oxidization wave outcompetes the reduction wave and pushes the counterline upward. The overall wave moves oppositely from bottom to top, forming a periodic exocytosis morphology. As the diffusive mass-transfer effect increases (Fig. 5 C and D) the deformation weakens for both $f=0.6$ and $f=1.0$, increasing the resistance to hydration effects and therefore its adaptability. At even larger $D_u=100a^2/\tau$, the mechanical relaxation time is longer than the diffusion time scale, and the shells converge to the homogeneous state.

Finally, we investigate the coupling strength ($\alpha$) on the shell morphology. For large values of $\alpha$ a bowl-like shape is observed throughout the chemical cycles without recovering its initial spherical morphology. We hypothesize that this is due to the high penalty of volume change assigned to the shell (SI Appendix, section S3C). As the shell swells, larger pores are expected that can temporarily increase the shell permeability to the solvent (47). However, due to the complexity of the problem, the changes in permeability are not considered in this work.

Discussion

In this paper we study autonomous gel membrane dynamics in closed shells triggered by periodic reduction–oxidation reactions. In particular, we focus on autonomous reduction–oxidation reactions where one of the components is immobilized into the shell modifying the microenvironment allowing for swelling and deswelling responses.

Our model is an extension of the reference metric remodeling approach for thin elastic plates (25, 26) to account for chemomechanical feedback loops. Using the two-fluid model approach (5–7) we relate the target metric changes to the response of polymer gels to chemical reactions. Using this approach we find that experiments in polymeric shells (14) can be explained by a small variation in the gel thickness (Fig. 3). In addition, we incorporate the chemical concentration changes due to mechanical deformations. We note that this approach differs from other thin-shell models (48, 49) that do not account for the geometrical feedback into the chemical response. However, the three-dimensional membrane gel model does consider mechanical feedback in hydrogels (50, 51). We find that mechanochemical feedback loops are important to generate autonomous patterns (Fig. 4).

Finally, we investigate future material design by coupling the material properties to the chemicals. We find that in silico multiresponsive materials can generate a wide variety of shapes in response to mechanical perturbations through changes in the chemical concentrations (Fig. 5 C and D). Our work inspires the design of mechanical stimulated materials (52, 53) and responsive shells to enhance the functionality of microcompartments (54) and of nanoreactors (55) that enhance catalytic activity. For example, one could in principle design catalytic microcompartments that can expand and contract to absorb or release components at a specific frequency that is favorable for processes that happen inside of the microcompartment.

Materials and Methods

Discrete Model of Thin-Shell Elasticity.

We discretize the shell to triangular lattice with the stretching energy written as

$$E_s=\frac{1}{8}\sum _T\frac{Eh}{1-{\nu }^2}\sqrt{\overline{g}}\left[{\left(Tr\left[{\overline{g}}^{-1}g-I\right]\right)}^2+2\left(\nu -1\right)\text{det}\left[{\overline{g}}^{-1}g-I\right]\right],$$ [5]

where $\overline{g}$ ($g$) is the target (actual) metric of the discretized triangle and $I$ is the identity matrix. $E$ and $h$ are the three-dimensional Young’s modulus and thickness of the shell, respectively, and $\nu =\frac{1}{3}$ is the Poisson ratio. For a homogeneous growing shell, the stretching energy and force can be estimated with the mesh triangle angles preserved where $E_s^h=\frac{1}{4}\frac{Eh}{1-\nu }{\sum }_T\frac{{\mathrm{\Delta }A}^2}{A_g}$ and $F_s^h=\frac{Eh}{\nu -1}\frac{4\pi }{N_v}\frac{r^2-r_g^2}{r_g^2}\stackrel{\rightharpoonup }{\mathbf{r}}$, with $N_v$ the total mesh vertex number and $r$ ($r_g$) the actual (target) radius of shell (SI Appendix, section S3A). With stretching energy dominant, the shell is relaxed to release the stress with relaxation time $\frac{\left(1-v\right)\gamma N_v}{8\pi Eh}$, which is inversely proportional to the Young’s modulus.

The bending energy can be discretized using Itzykson discretization (56), with

$$E_b=\sum _j\left(\frac{1}{2}{\kappa }_j{\left(H_j-2H_0\right)}^2+{\kappa }_j^g{G}_j\right)\hspace{0.17em}{A}_j,$$ [6]

where $H_j=\frac{\text{sign}}{A_j}\left|\frac{1}{2}{\sum }_T\left(\text{cot}{\theta }_l{\mathbf{r}}_{jk}+\text{cot}{\theta }_k{\mathbf{r}}_{jl}\right)\right|$ and $G_j=\frac{1}{A_i}\left(2\pi -{\sum }_T{\theta }_i\right)$ are the mean curvature and Gaussian curvature of vertex $j$, respectively. $A_j=\frac{1}{8}{\sum }_T\left(\text{cot}{\theta }_l{r_{jk}}^2+\text{cot}{\theta }_k{r_{jl}}^2\right)$ is the dual-lattice area of vertex $j$ and $\kappa =\frac{1}{12}\frac{E{h}^3}{1-{\nu }^2}$ and ${\kappa }^g=\left(\nu -1\right)\kappa$ are the bending rigidity and Gaussian bending rigidity correspondingly. $T$ is the neighbor triangles of vertex $j$ and ${\theta }_j$ is the angle of vertex $j$ on triangle $T$.

Chemical Reaction.

The chemical reactions (Eq. 2) are evolved on the shell by explicit time integration following the procedure

$$U^{n+1}=A_c\left(U^n+dt{D}_U{\mathrm{\Delta }}_M{U}^n+dtF\left(U^n\right)\right),$$ [7]

where the Laplace–Beltrami operator (${\mathrm{\Delta }}_M$) is implemented through the discrete exterior calculus method (57) and the chemical concentration is rescaled by the vertex area fraction $A_c={\sqrt{g}}^n/{\sqrt{g}}^{n+1}$ (58) (see SI Appendix, section S2 for details). $D_U$ is the diffusion coefficient of chemical $U$. At each time step, we couple the target metric to the catalyst concentration ($v$) with ${\overline{g}}_{\alpha \beta }=\left(1+\alpha \hspace{0.17em}v\left(r,t\right)\right){\overline{g}}_{\alpha \beta ,t=0}$ following the fact that the target metric is modified subject to the hydrophilicity change of polymers, where $\alpha$ is the coupling coefficient depending on the intrinsic properties of the hydrogel. A detailed discussion about the growth ratio is shown in SI Appendix, section S4 using gel theory, which is linearly dependent on the catalyst concentration at a small amount of chemicals changes.

In the simulation system, the BZ chemical reaction is measured in units of $\tau =1/\left(k_3\left[A\right]\right)$, where $k_3$ is a rate constant and $\left[A\right]$ is the concentration of a chemical substrate. The thickness and radius of the shell are measured in units of $a$ and the elastic relaxation has time unit ${\tau }_{elastic}=\gamma /\left(Ea\right)$, which is significantly small compared to the chemical reaction. The quantity and units of the other parameters are shown in SI Appendix, Table S2.

Data Availability

All study data are included in the article and/or SI Appendix.