## Abstract

Spatially periodic, temporally stationary patterns that emerge from instability of a homogeneous steady state were proposed by Alan Turing in 1952 as a mechanism for morphogenesis in living systems and have attracted increasing attention in biology, chemistry, and physics. Patterns found to date have been confined to one or two spatial dimensions. We used tomography to study the Belousov-Zhabotinsky reaction in a microemulsion in which the polar reactants are confined to aqueous nanodroplets much smaller than the scale of the stationary patterns. We demonstrate the existence of Turing patterns that can exist only in three dimensions, including curved surfaces, hexagonally packed cylinders, spots, and labyrinthine and lamellar patterns.

In his seminal paper, Alan Turing (*1*) aimed to provide a mechanism for self-regulated pattern formation in biology by showing that sets of reaction-diffusion equations with appropriate kinetics and diffusion coefficients could spontaneously evolve to spatially periodic structures. The main requirements for Turing-pattern formation in a reaction-diffusion medium are long-range reaction inhibition (fast diffusion of an inhibitor) and short-range activation (slow diffusion of an activator) (*1*). The first experimental demonstration of a Turing pattern (*2*, *3*) occurred in the chlorite–iodide–malonic acid (CIMA) reaction (*4*), in which complexation of the activator species (iodine) with the indicator (starch) generated the necessary separation between the effective diffusion constants of activator and inhibitor. Recent advances in molecular biology have made it possible to establish the Turing mechanism in several biological systems (*5*–*7*), but many important aspects of the reaction-diffusion processes in such systems remain to be clarified (*8*, *9*). One key and still largely unexplored aspect affecting Turing-pattern formation is dimensionality; that is, how the number of spatial dimensions influences the pattern behavior. Actual patterns can never be truly planar [two-dimensional (2D)] at a molecular level, as is often assumed. This 3D-to-2D simplification is adequate only when the characteristic Turing wavelength (*l*_{T}) of the pattern is sufficiently (at least two times) greater than the thickness of the medium; for example, in the case of skin patterns (*9*, *10*). If the linear size of the system in all three dimensions exceeds *l*_{T}, then 3D Turing patterns may differ from the well-studied 2D patterns.

The issue of possible differences between 2D and 3D patterns was raised for the first time in connection with some of the earliest experiments on 2D Turing patterns (*2*, *3*). A number of theoretical works predict a variety of possible shapes for 3D patterns (*11*–*13*)—including body-centered cubic (bcc) arrays of spots, hexagonally close-packed (hcp) cylinders), and lamellae—that are not simply extensions of 2D patterns into 3D. However, experiments in continuously fed unstirred reactors (CFURs) on the CIMA system or others (*14*) that are able to generate Turing patterns have been effectively confined to 2D patterns in a thin layer or to 3D patterns in a thick layer with large concentration gradients (i.e., an inhomogeneous system) (*15*). Moreover, the CFUR configuration does not lend itself to the analysis of 3D patterns (*16*) by tomographic methods, which have successfully been applied to the study of 3D scroll waves (*17*–*19*), a straightforward extension of 2D spiral waves into 3D media. Visualization of 3D patterns is a challenging task; simple visual observation or microscopy yields a 2D projection, which will obscure the structure in all but the simplest patterns.

To generate 3D Turing patterns in a homogeneous medium while simultaneously overcoming the technical problems that hinder their observation, we explore the Belousov-Zhabotinsky (BZ) reaction incorporated into a water-in-oil aerosol OT (AOT) microemulsion, the so-called BZ-AOT system (*20*, *21*). This medium has already been shown to give rise to 2D Turing patterns that are stable for 1 to 2 hours without external feeding (*22*). The ~10-nm–diameter aqueous droplets of the microemulsion are surrounded by a monolayer of surfactant, sodium bis(2-ethylhexyl)-sulfosuccinate and are dispersed in a continuous oil phase. The polar BZ reagents (malonic acid, sodium bromate, sulfuric acid, and ferroin) reside in the water droplets. When the reaction begins, apolar intermediates (notably the inhibitor species, Br_{2}) are produced in the droplets and diffuse through the oil phase. The diffusion of Br_{2} occurs at rates up to two orders of magnitude greater than that of the polar species, including the BZ activator HBrO_{2}, which diffuse together with entire water droplets.

Note that the microheterogeneous BZ-AOT system can be considered as homogeneous on the chemically relevant length scale, because the characteristic wavelength of the patterns observed (*l*_{T} = 0.1 to 0.3 mm) is several orders of magnitude greater than the size and spacing of the of water droplets (~10 nm). The characteristic times of the key chemical processes in the BZ reaction, typically on the order of seconds, are also much longer than the characteristic time for mass exchange via fission-fusion of water droplets.

We used BZ reactant concentrations and microemulsion parameters [the water-surfactant molar ratio ω = [H_{2}O]/[AOT] (where [H_{2}O] and [AOT] are the concentrations of H_{2}O and AOT, respectively) and ϕ_{d}, the volume fraction of droplets] that give rise to Turing patterns in 2D. To improve visualization, we replaced the octane used as the oil phase in earlier experiments with cyclooctane, which requires a small readjustment of the concentrations to remain in the Turing domain. Cyclooctane has nearly the same index of refraction as quartz, which minimizes refraction at the curved interfaces between the capillary and its surroundings. For the 2D experiments, we used a protocol described earlier (*22*), in which we sandwiched the BZ-AOT emulsion between two optical windows separated by an 80-μm-thick polytetrafluoroethylene (Teflon, DuPont, Wilmington, Delaware) gasket. Patterns were observed via a microscope equipped with a charge-coupled–device camera. Stationary Turing structures emerged after initial transient bulk oscillations and/or waves and persisted for 1 hour or more. At longer times, we observed changes (for example, from labyrinthine or lamellar to spot patterns) and, ultimately, the disappearance of structured patterns.

We conducted the 3D experiments in a cylindrical quartz capillary with an inner diameter of 0.3 to 0.6 mm (i.e., several *l*_{T}). The capillary was filled with the BZ-AOT mixture, sealed at the bottom, and vertically submerged in a rectangular cuvette containing cyclooctane, the oil used in the BZ-AOT medium. Visualization of the 3D Turing structures was then implemented by optical tomography performed on a series of absorption images taken within a full rotation of the capillary. The samples were typically rotated at 0.25 revolutions per second with a frame capture rate of 20 frames per second. Concentration fields (oxidized form of the catalyst) were obtained by inverse Radon transformation of the filtered image data (*23*). The 2D and 3D stationary structures were moderately stable to vibration and shear stress caused by starting the rotation motor or removing the sample from the holder. More details are available in the supporting online material. We obtained 2D and 3D patterns in experiments using microemulsions with the same composition.

In 2D, stationary Turing structures generally consist of imperfect hexagonal arrays of spots or labyrinthine arrangements of stripes (Fig. 1, A and D). In 3D, we observed a variety of new patterns (Fig. 1, B, C, and E to G). When the BZ-AOT system was in the central portion of the 2D spot region in the space of initial concentrations (Fig. 1A), we also observed spots in 3D (Fig. 1B). A 3D reconstruction of the patterns seen in Fig. 1B is presented in Fig. 2A. As we changed the initial concentrations to move closer to the boundary between the 2D spot and stripe regions, we observed long, hexagonally packed cylinders in 3D; i.e., an hcp pattern (pattern in Fig. 1C and its 3D reconstruction in Fig. 2, B and C). In the parameter regime of 2D labyrinthine stripes (as in Fig. 1D), the 3D configuration yielded labyrinthine patterns (Fig. 1E and Fig. 2D), closed tubular surfaces (Fig. 1F and Fig. 2E), and half-pipe–shaped structures (Fig. 1G and Fig. 2F).

Note that such patterns as hcp (Fig. 2B) and stationary surfaces (Fig. 2, E to H) can exist only in 3D. Curved surfaces appear to be favored by narrow capillaries, where the influence of the lateral boundary is strong. The labyrinthine structure in Fig. 2D is interconnected, a feature clearly visible in a capillary with larger diameter. The patterns in Fig. 2, D to F, were obtained at the same average initial concentrations, but at slightly different local initial conditions, and these distinct patterns can coexist in the same capillary but in different regions along its length. Differences in initial conditions resulted from the waves initially appearing in the capillary before the Turing patterns arose. These waves generated an inhomogeneous concentration distribution from which the stationary Turing patterns gradually emerged.

Although these patterns are stationary for a time that corresponds to dozens of cycles of the BZ-AOT system under oscillatory conditions, our patterns were necessarily transient because the system was closed. In an effort to better understand our experimental results and to test whether these patterns could indeed be stable, we performed 3D simulations using a two-variable (*x*, *z*) Oregonator-like model of the BZ-AOT system that is very good at reproducing many of the 2D structures observed in the BZ-AOT system (*24*). In Fig. 3, A and B, we present 3D spots and hexagonal close-packing obtained at model parameters that give spots in 2D. As in the experiments, hexagonal close-packing (Fig. 3B) emerged when the system was near the boundary in parameter space between the 2D spot- and stripe regions; patterns corresponding to the experimentally observed patterns in Fig. 2, D to F, are shown in Fig. 3, C to E. In agreement with the experiments, these patterns were obtained with model parameters (the same for all patterns) at which stripes emerged in 2D. The simulations differed only in the initial distribution of *x*: random for labyrinthine (Fig. 3C), radially symmetric Gaussian in the horizontal plane for tubes (Fig. 3D), Gaussian symmetric with respect to a vertical plane through the cylinder axis for lamellae (Fig. 3F), and asymmetric (off-center radially symmetric Gaussian) for the half-pipe (Fig. 3E).

Our results demonstrate the potential importance of dimensionality and initial conditions for pattern formation in nature. Pattern selection is circumscribed by the number of available spatial dimensions, and the characteristics of a stationary pattern in a given region of the system are determined by the local conditions present at the early stages of pattern formation. The possibilities for spatial multistability are considerably greater in three dimensions than in two, and pattern selection via gradients established during early stages of development seems an eminently plausible scenario in living systems. Theorists predict many other 3D Turing patterns, such as bcc, face-centered cubic, or perforated lamellae (*25*). Experimental detection of such patterns—which depend, among other factors, on whether the Turing instability is sub- or supercritical—may be expected.

Biological Turing patterns in 3D have not yet been reported, even though there has been considerable effort to find Turing-driven morphological phenomena. With the growing interest in Turing’s theory as a model for morphogenesis (*1*) and emerging examples of 2D Turing patterns in living systems—for example, disposition of feather buds in chicks (*6*) and hair follicles in mice (*7*)—the near future seems likely to unveil evidence of 3D self-regulated patterns in living systems. Candidates include skeletal pattern formation in developing chick limbs (*26*) and the head regeneration process in *Hydra* (*27*), a species that inspired Turing in developing his groundbreaking theory of morphogenesis. In addition, 3D Turing patterns [especially those in the subcritical regime, where 3D localized patterns can exist (*28*)] may be useful vehicles for information storage (*29*).

## Supporting Online Material

www.sciencemag.org/cgi/content/full/science.1200815/DC1

Materials and Methods

Figs. S1 and S2

## References and Notes

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- We thank M. Hauser for helpful discussions, J. Carballido-Landeira for preliminary experiments, and F. Mello for assistance in constructing the tomography apparatus. This work was funded by the NSF under grants CHE-0615507, CHE-0526866, and NSF Materials Research Science and Engineering Center grant DMR-0820492 and a U.S.-Hungarian Cooperative Grant. I.R.E. thanks the Radcliffe Institute for a fellowship.