Role of Nuclei in Liesegang Pattern Formation: Insights from

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Role of Nuclei in Liesegang Pattern Formation: The Insights from Experiment and Reaction-Diffusion Simulation Masaki Itatani, Qing Fang, Kei Unoura, and Hideki Nabika J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b12688 • Publication Date (Web): 26 Jan 2018 Downloaded from http://pubs.acs.org on January 30, 2018

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The Journal of Physical Chemistry

Role of Nuclei in Liesegang Pattern Formation: The Insights from Experiment and Reaction-Diffusion Simulation Masaki Itatani,1 Qing Fang,2 Kei Unoura,3 Hideki Nabika3* 1

Graduate School of Science and Engineering, Yamagata University, 1-4-12 Kojirakawa,

Yamagata 990-8560, Japan, 2Department of Mathematical Science, Faculty of Science, Yamagata University, 1-4-12 Kojirakawa, Yamagata 990-8560, Japan, 3Department of Material and Biological Chemistry, Faculty of Science, Yamagata University, 1-4-12 Kojirakawa, Yamagata 990-8560, Japan

* [email protected]

ACS Paragon Plus Environment

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ABSTRACT Many types of periodic patterns can spontaneously form in nature across wide spatiotemporal scales. Construction of chemical model that mimic these periodic patterns are of considerable interest from both scientific and technological viewpoints. The Liesegang phenomenon is one of the chemical models to form periodic patterns with well-defined periodicity. However, the parameters that influence the mechanism and resultant pattern geometry are not completely known. In this study, we use surface chemistry methods to evaluate the influence of nucleation threshold on the geometry of Liesegang patterns. Cysteine was used as an additional ligand for the precursors (Agn nuclei and/or Ag nanoparticles in the present system) to reduce their surface free energy and thus the nucleation free energy. As a result, the formed Liesegang patterns had smaller spacing coefficient (i.e., finer periodic patterns), a phenomenon that was also reproduced using reaction-diffusion simulation with lowered nucleation threshold. The small spacing coefficient at lowered nucleation threshold was discussed in terms of a slower rate of Ostwald ripening. Similar control of the nucleation threshold through surface chemistry can be applied to various precipitation systems, as well as gaining insight into the comprehensive mechanism underlying various animate and inanimate patterns formed in nature.

INTRODUCTION The formation of spatiotemporal patterns at conditions far from equilibrium is ubiquitous in nature, and it plays a critical role in the dynamic, diverse, flexible, and robust features of both animate and inanimate matters.1 Spatial patterns are often discussed in terms of reaction and diffusion (RD) phenomenon caused by cooperative interaction between reaction and diffusion of


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substances in the system. There are two types of spatial patterns: traveling patterns and static patterns. Nerve impulse propagation in animals is a typical traveling wave. It is dominated by the activity of ion channels and diffusion of Na+ and K+ ions, and is well explained with the FitzHugh-Nagumo model.2 When the biological activity and diffusivity are replaced by chemical reactivity and diffusivity, respectively, chemical traveling waves are formed in vitro, such as the Belousov-Zhabotinsky (BZ) reaction and proton waves.3–9 Since the key parameters that determine the propagation characteristics can be easily tuned by experimental conditions, these chemical waves are used as models to gain physicochemical insight into traveling waves. Static patterns, on the other hand, can be seen in various fields including chemistry,10,11 biology,12,13 and bacteriology.14,15 Similar to the traveling waves, static patterns formed in various animate and inanimate systems can be modeled by chemical reactions. For example, the stripe pattern of zebrafish can be reproduced by chemical reactions with the Turing mechanism that produces various patterns from an initially homogeneous medium.16,17 Thus, revealing the mechanism of pattern formation in chemical reaction systems can help us understand the rich features seen in natural patterns. While the Turing patterns are formed in a homogeneous medium, the pattern formed in the presence of a concentration gradient in one reactant is known as the Liesegang pattern.18 In chemical experiments, the gradient can be formed by a contact between a gel containing an inner electrolyte and an aqueous solution containing an outer electrolyte. From the solution-gel interface, the outer electrolyte diffuses into the gel media and forms a concentration gradient. When the reaction between the inner and outer electrolytes gives insoluble products, the products will deposited as a precipitate and form periodic precipitation bands in the gel media. There are various reports on chemical Liesegang patterns that are formed by sparingly soluble salts,19–25


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chemical reduction,26 and polymerization,27 where the precipitate could be hydroxides,28–34 chromates,19–25 halides,35,36 carbonates37 and metal nanoparticles,26 polymers,27 hydrogels,38,39 and so on. Although both Turing and Liesegang patterns are periodic and formed spontaneously, their periodicity is substantially different. In the Turing pattern, the inter-band spacing is constant, and this distance depends on the diffusivity and the period of the limit cycle.40,41 In contrast, the inter-band spacing of Liesegang pattern varies as a function of distance from the interface.20,22,26,27,39 Thus, the Turing and Liesegang mechanisms, both appearing in different animate and inanimate systems, are important for understanding pattern formation in nature. The Turing pattern has been comprehensively examined by interdisciplinary studies spanning chemistry, biology, physics, and mathematics. In contrast, for the Liesegang pattern, there is still a large gap between nature and existing chemical models. Several patterns that are similar in geometry to the Liesegang patterns have been reported in bacteria colonies,14 biological tissue,42 rock,43 and the solar system. However, there is no direct evidence to link these natural patterns to the chemical models of Liesegang pattern. One problem lies in the difficulty of controlling the reaction condition for Liesegang pattern formation in chemical systems. Many proposed models involve a nucleation process followed by precipitate growth, such as in the nucleation and growth44–47 and sol-coagulation models.22,45,48 For example, the nucleation and growth model involves the following three processes: Reaction: A + B → C

(1)

Nucleation: C → N , if [C] > C*

(2)

Precipitate growth: N → P (3)


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where A, B, C, N, and P are the outer electrolyte, inner electrolytes, reaction product, nuclei of particles, and precipitate, respectively. In this model, A diffuses into the gel medium containing B and reacts with B to form C (eq. 1). In the gel medium, C nucleates and forms N if local concentration reaches the nucleation threshold C* (eq. 2), which will be followed by spontaneous nuclei growth and precipitation (eq. 3). From this model, it is clear that C* controls the progress of precipitate and band formation. Indeed, recent studies based on RD simulation demonstrated the critical importance of C* on the geometry of Liesegang patterns.49–51 C* represents the energy barrier (ΔG) for the transition of the product from the molecular state (C) to solid state (N) with non-zero surface area and volume, as defined by the following equation including two terms corresponding to surface and volume effects:52 ΔG = -4/3πr3ΔGv + 4πr2γ

(4)

where r, ΔGv, and γ are the radius of nuclei, the free energy per unit volume of nuclei or nanoparticles, and the surface free energy per unit area, respectively. This equation implies that C* can be controlled through ΔG, in which ΔGv and γ would play critical roles. However, there is little knowledge about the relation between Liesegang pattern formation and either ΔGv or γ, except an experimental study that used impurities to control the number of nucleation sites in the reaction medium.22 In that research, the authors clarified that the addition of impurity led to heterogeneous nucleation that was discussed by varying C* in their model. However, the relationship between ΔG and C* is still unknown, even though it provides an alternative and flexible approach to control the precipitation geometry. Here, we used a new experimental approach to understand the chemical Liesegang patterns through direct control of C* with the aid of surface chemistry. Surface chemistry has


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already been used to control the γ value of nanomaterials (such as metal clusters and nanoparticles) that are the dominant chemical species during a nucleation and precipitation growth processes.53 For this purpose, we chose a system based on the chemical reduction of Ag+ that yields Liesegang patterns consisted of Ag nanoparticles,26 because the γ value of Agn clusters can be easily tuned by various ligands such as thiols,54–59 phosphate,60,61 and citrate.62–64 Since Agn would be the transition species from Ag atoms to Ag nanoparticles, the control of their γ value may clarify the relationship between Liesegang pattern formation and C* or ΔG. In the present study, we used cysteine as an additional ligand for the stabilization of Agn species, because cysteine is easily dissolved in a gelatin medium and expected to strongly ligate Ag species through its sulfide and amino groups. As a result, we found that the presence of cysteine has a strong influence on the geometry of the Liesegang pattern, which was discussed from RD simulation with varying C*. Furthermore, in order to eliminate the ligation effect of citrate as a reducing agent,26 we also carried out similar experiments with cysteine but no citrate in a photochemical reduction, instead of the chemical reduction with citrate. The results from chemical and photochemical experiments, along with RD simulations, revealed that the ligation of cysteine to Agn decreased C* due to a reduction in γ. As a result, the geometric characteristics of the formed Liesegang pattern were changed. This study, based on surface chemistry, therefore unlocks a new approach to gain deep insight into how chemical conditions control the pattern geometry in the Liesegang mechanism.

EXPERIMENTS Reagents.


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Gelatin (fine powder) and sodium sulfate were purchased from Nacalai Tesque Industry. Silver nitrate, trisodium citrate dehydrate, L-cysteine, and agarose were purchased from Wako Pure Chemical Industry. All reagents were used without further purification.

Chemical reduction system. A gelatin solution (5.0%) was prepared with Milli-Q-water under stirring and heating at 75 ℃ for 25 min. Sodium sulfate and trisodium citrate dihydrate were added to the gelatin solution to be final concentrations of 130 mM and 45 mM, respectively. For the system with cysteine, L-cysteine was also added to the gelatin solution at 50 μM. The mixed gelatin solution was stirred and heated at 90 ℃ for 25 min. Then, 1.5 mL of the solution was poured into glass petri dish (inner diameter: 59 mm) and stored in an incubator at 18 ℃ overnight to yield the gelatin gel matrix. An agarose solution (8.0%) was prepared with Milli-Q-water, and degassed in vacuum for 5 min. Then, the solution was heated by microwave for 40 s. Immediately after heating, this solution was poured into a silicon tube (inner diameter: 7 mm) and left for 20 min at room temperature. Then, agarose stamps were made by slicing the agarose gel into columns with length of 3 cm, and soaked in silver nitrate aqueous solution (1.0 M) for more than 2 weeks. The agarose stamp doped with Ag+ ion was then put on the surface of gelatin gel formed in the petri dish, and the stacked gels were stored at 18 ℃ for 6 h. Since gelatin plays an important role to control the reduction and nucleation rate of silver species, we used the gelatin gel as the reaction medium. On the other hand, the agarose gel was used as the stamp because of


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its easily usability compared with the gelatin gel.26 The obtained patterns were observed using an optical microscope BX-43 (Olympus Co., Ltd., Japan).

Photochemical reduction system. The photochemical reduction system was prepared in the same way as for the chemical reduction system, except trisodium citrate dihydrate was not used. The agarose stamp was put on the surface of gelatin gel in petri dish, which was then transferred to a thermoplate and kept at 24 ℃. Then, the photochemical reduction was carried out by irradiating UV light (250–400 nm, 45–50 lx; MAX-301, Asahi Spectra Co., Ltd, Japan) for 4 h on the thermoplate. The obtained patterns were observed using the same optical microscope.

RESULTS AND DISCUSSION In the chemical reduction system without cysteine, periodic brown bands appeared spontaneously, in which multiple band patterns could be observed outside the continuous precipitate (Fig. 1a). The brownish precipitates are silver nanoparticles from the reaction between Ag+ and citrate. A previous work showed that two different Ag+ species exist in the presence of gelatin molecules:65 freely diffusing Ag+ ion (free Ag+) and Ag+ ion ligated to amino acid residues in the gelatin matrix (bound Ag+). It has been suggested that only the bound Ag+ is involved in Liesegang pattern formation.26 Since the bound Ag+ and bound Ag0 from reduction were relatively stabilized by the amino acid residues, nucleation commenced when the local concentration of the bound species reached the threshold of C*.65 Afterwards, Ag nanoparticles


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were formed via particle growth and Ostwald ripening, producing the brownish color in Fig. 1a owing to surface plasmon resonance. Since Ag+ diffused from the agarose stamp according to the Fick’s law, the brownish precipitate appeared and extended from the agarose stamp with time (Fig. 1b). Just after placing the agarose stamp on the gelatin gel (0 min), the gelatin gel was colorless with no visible precipitations. At 60 min, the reaction front can be seen on the left side, forming precipitates as a brownish band. This precipitation zone progressed with time, and the reaction front reached the center of the imaging area at 180 min, at in which point continuous precipitation changed to discrete periodic precipitation at the edge of the reaction front. More obvious periodic precipitation was observed at 360 min. The appearance of periodic patterns was also shown as an oscillation of intensity in the line profile (Figure 1c). The inter-band distance was about 100–200 m, and this distance increased further away from the edge of agarose stamp. In the presence of cysteine, similar precipitation propagation also occurred, but the periodicity was different. In Fig. 2a, it appears that only a continuous precipitation zone was formed, unlike the periodic bands at the reaction front when cysteine was not used (Fig. 1a). The enlarged image in Fig. 2b also showed no periodic bands with the inter-band distance of 100–200 m. However, this system did yield bands but with much finer periodicity. At 60 min, fine and faint bands started to emerge from the left side, which can be clearly seen in the high-contrast images shown in Supporting Information Fig. S1 and the intensity line profile (Fig. 2c). Such fine patterns were not observed in the absence of cysteine, even when high-contrast images were examined. As the reaction proceeded, the dark continuous precipitation band on the left expanded and became visible after 180 min, preceded by the fine bands. The inter-band spacing of this fine pattern was below 100 m, meaning that its periodicity was almost half that of the pattern formed without cysteine. Therefore, it can be concluded that the presence of cysteine


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affected the chemical condition for the periodic band formation, presumably through the ligation to Ag species such as Agn nuclei and Ag nanoparticles and thereby altering their surface free energies (γ). Now we consider the mathematical model of reaction-diffusion (RD) simulations. The diffusion part of a chemical molecule u is described by div(D(𝜏,x) grad u). When D(𝜏,x) is homogeneous, the diffusion part is given by 𝐷∇2 𝑢 = D div(grad u). The reaction process is modeled by a dynamical system 𝑢𝜏 = F(𝜏,x, u). In order to discuss the role of cysteine, RD simulations were carried out based on the nucleation and growth model66 as follows: ∂𝐴 ∂𝜏

= ∇2 A －R (A, B)

∂𝐵

𝐷

= 𝐵 ∇2 B －R(A, B) ∂𝜏 𝐷 𝐴

∂𝐶 ∂𝜏

(6)

𝐷

= 𝐷𝐶 ∇2C +R(𝐴, 𝐵)－C𝜃(𝐶－𝐶 ∗ )－CN(C, D)

∂𝐷 ∂𝜏

(5)

𝐴

= C𝜃(C－𝐶 ∗ ) + CN(C, D)

(7)

(8)

where A, B, C and D are the concentrations of the outer electrolyte, inner electrolyte, reaction product, and precipitate, respectively. The boundary conditions for A, B and C are homogeneous Neumann boundary conditions. 𝜏 is the dimensionless reaction time (𝜏 = 𝑡𝐷𝐴 /𝐿2 ), where L is the characteristic size of the pattern. ∇2 is the Laplacian operator. 𝐷𝐴 , 𝐷𝐵 and 𝐷𝐶 are the diffusion coefficients of A, B and C, respectively. R is the reaction kinetics function. The function N(C, D) describes the dependence of the precipitation pathway on the presence of pre-existing precipitates at the grid point of interest. The inner electrolyte B is homogeneously distributed in


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the calculation grid, whereas A is initially present in a circular region at the center of the calculation grid. The parameters used in the present simulation are: A0 = 1.0, B0 = 0.01, C0 = D0 = 0, and 𝐷𝐴 ∶ 𝐷𝐵 ∶ 𝐷𝐶 = 1.0 : 0.75 : 0.1. The five-point formula with space grid-step Δx = 0.25 is employed on a 1600 × 1600 grid, and the fourth-order Runge-Kutta method with time step Δt = 0.01 is used to integrate the semi-discretized ordinary differential equations. The 𝜃(C－𝐶 ∗ ) is a step function: 𝜃(C－𝐶 ∗ ) = 1 when C－𝐶 ∗ ≥ 0, while 𝜃(C－𝐶 ∗ ) = 0 when C－𝐶 ∗ < 0. Thus, the value of C* is very important because it controls the progress of precipitate formation (eq. 8). To evaluate the effect of cysteine in modifying γ due to the ligation to Agn nuclei, we conducted the RD simulation by changing the value of C*, because eq. 4 implies that γ alters C* through ΔG, as discussed earlier. Fig. 3 shows the RD simulation results with different C*, all of which successfully produced periodic patterns similar to the ones experimentally observed. At C* = 0.011, relatively thick bands formed a concentric pattern. By decreasing C* to 0.015, each band became thinner, and the concentric patterns became denser. Further decreasing C* to 0.01 made the bands much thinner and almost linear, and multiple band patterns were aligned in a circular configuration. To compare the geometrical characteristics of experimental and simulated patterns, we analyzed the spacing law for the Liesegang phenomenon, in which the ratio between the positions of two consecutive bands (xn+1/xn) tends to converge to a constant known as the spacing coefficient (1 + p) as n becomes large. The experimental spacing coefficients with and without cysteine are shown in Fig. 4a, where n = 1 is defined as the innermost band that could be resolved under the optical microscope. In other words, the absolute value for n has less meaning than the relative order among the resolvable bands. Experimentally, the spacing coefficient converged to a constant value in each case, but the values are different (1.05 without and 1.02


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with cysteine). The smaller spacing coefficient in the latter means the formation of finer periodic bands in the presence of cysteine (Fig. 2). A similar tendency was seen in the RD simulations, i.e., the spacing coefficient decreased with decreasing C* (Fig. 4b). However, unlike the experimental results, the simulated spacing coefficient showed a gradual decrease at small n. A possible reason is that the simulation predicted quite thin precipitation bands that could not be resolved under the experimental observation. Thus, the simulated bands in Fig. 3 would contain much thinner bands than the experimental ones in Fig. 1 and 2. After this gradual decrease, the simulated spacing coefficient became constant at larger n. Although the spacing coefficients with different C* appear to converge to the same value, an enlarged plot (inset in Figure 4b) shows a significant difference among them under the observation condition. Therefore, the simulation result suggested that a reduction in the spacing coefficient, which was observed experimentally by adding cysteine, was induced by a decrease in C*. Thus, it was demonstrated that the addition of cysteine decreased C* through its ligation to Agn nuclei, which led to the formation of fine bands in both experiments and RD simulations. From these results, we concluded that the addition of cysteine produces a similar effect to that of reducing C* in the pattern formation. The added cysteine can reduce the surface free energy γ of Ag clusters by stabilizing their surface atoms. A lowering γ means a reduced energy barrier (ΔG) for the transition from the molecular state (bound Ag) to solid state (Ag nuclei or Ag clusters) with non-zero surface area and volume (eq. 4). Then, nucleation can proceed under a lower excess concentration C* of the molecular species. Thus, it can be concluded that the presence of cysteine can help to lower C*, as similarly observed in the spacing coefficients from experiments and simulations. In the simulation, the width of each band and inter-band spacing narrowed when C* was decreased from 0.011 to 0.01, in accordance with our experimental


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results. In this way, the formed bands shrank and finally became linear ones, and the inter-band spacing also decreased upon the addition of cysteine. This agreement strongly suggests that the addition of cysteine contributed to the lowering in C*. However, regardless of the addition of cysteine, gelatin gels used in these chemical reduction experiments contained citric acid as a reducing agent. Previous works have reported that citrate ligates to Agn clusters to form Agn-citrate complex,62–64 which can also modulate the ΔG for nucleation and affect the pattern formation process. Therefore, there remains the possibility that citrate and cysteine compete in ligating to Agn clusters, which makes the effect of ligand ligation on ΔG somewhat complicated. In order to clarify the effect of cysteine, a reducing agent that cannot act as ligand for Agn clusters should be considered. For this purpose, we also conducted photochemical reduction by irradiating UV light to reduce Ag+ ion, instead of using citrate.67,68 Fig. 5 shows the formed precipitation bands after UV irradiation for the citrate-free samples without and with cysteine. Both samples formed bright yellow (owing to surface plasmon resonance of Ag nanoparticles) precipitation bands around the agarose stamp, indicating that the reduction reaction proceeded under UV irradiation. The color of the precipitate looks different between the chemical and photochemical reduction (brown vs. yellow), and this was attributed to a difference in the duration of light exposure under the microscope to visualize the fine periodic patterns. Without cysteine (Fig. 5a), a yellow precipitation zone was formed around the stamp, and an orange band appeared at 2 mm away from the edge of stamp. The enlarged image and intensity line profile (Fig. 5c) showed that, in the system without cysteine, the orange band was homogeneous without fine periodic structures. In the presence of cysteine, the orange band contained concentric fine bands according to the enlarged image (Fig. 5b). These fine bands were confirmed to satisfy the same


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spacing law, and the value of 1+p converged to a constant (Fig. 5d). Between the photochemical experiments with and without cysteine, it became even more clear that surface ligation of Agn clusters reduces the inter-band distance and forms finer Liesegang patterns, which is likely through controlling the values of γ and ΔG for nucleation. From our experiments, we propose that the surface ligation lowers the spacing coefficient. C* represents the concentration at which the nucleation process occurs (eq. 2), which can also be defined as the maximum concentration of C that can be used for the nucleation and growth steps. Above C*, nuclei are formed and grow with Ostwald ripening (eq. 3), and they can exhibit periodic structure consisted of precipitation and depletion zones. Since the number density of nuclei is proportional to C* (because nucleation cannot start below this concentration), a lower C* means lower densities in the formed nuclei and particles. Furthermore, the rate of Ostwald ripening is proportional to the molar volume and thus the number of particles.69 Above all, lowering C* reduces the Ostwald ripening rate, and as a result thinner precipitation and depletion bands are formed. Therefore, in the suggested mechanism, surface ligation leads to finer Liesegang patterns through lowering the surface free energy. Since there are other possible ligands than citrate or cysteine, an appropriate design of ligands for the nuclei allows well-defined control of the geometry of Liesegang patterns. Furthermore, the relationship between ΔG for nucleation and the spacing coefficient found in the present study may help to reveal the formation mechanism of Liesegang-like patterns with diverse geometries in nature. Finally, our finding indicates that besides the chemical and physical characteristics of the precipitating materials, coexisting materials that can interact with the growing precipitate particles can be another key parameter to gain novel insight and comprehensively model the Liesegang phenomenon.


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CONCLUSION Our experiments and simulations showed that the surface free energy of precipitate nuclei affects the spacing coefficient in the resulting Liesegang patterns. Because cysteine as a ligand has a high affinity to Agn clusters, adding cysteine in the gelatin gel matrix caused the formation of finer periodic patterns, and the periodicity was confirmed to satisfy the spacing law for Liesegang phenomena. Similar results were observed in the corresponding photochemical reduction system, thereby eliminating the influence of citrate on the surface free energy. While our experiments showed that the addition of cysteine changed the geometry of the Liesegang pattern, the RD simulation suggested that such fine structure could also be observed when C* became lower. All these results indicate that surface ligation of cysteine to Agn clusters reduces the surface free energy and C*, resulting in the appearance of finer periodic patterns due to the slower rate of Ostwald ripening. Such control of C* via γ could be applied to various other precipitation systems. Furthermore, there are many choices of ligands to modulate γ. Thus, this surface chemistry-based approach to explore Liesegang phenomena is useful for understanding the overall mechanism underlying the formation of patterns in various animate and inanimate natural systems.

Supportring Information. High contrast images of reaction fronts.


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ACKNOWLEDGMENT

This work was supported by JSPS KAKENHI Grant Number 16H04092.

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(a)

4 mm

(b)

(c) 0 min

0 min

60 min

60 min

180 min

180 min

360 min

360 min

1 mm

Figure 1. Liesegang patterns obtained in the chemical reduction system without cysteine. (a) Microscope image of sample 6 h after the agarose stamp (disk at the center of image) was placed on the gelatin gel. (b) Enlarged and successive images of the propagating precipitation reaction front, for the area shown in white rectangle in (a). (c) Line profiles of the intensity in images shown in (b). The black arrows at 180 and 360 min indicate the positions of individual bands.


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(a)

4 mm

(c)

(b)

0 min

0 min

60 min

60 min

180 min

180 min

360 min

360 min

1 mm

Figure 2. Liesegang patterns obtained in the chemical reduction system with cysteine. (a) Microscope image of sample 6 h after the agarose stamp was placed on the gelatin gel. (b) Enlarged and successive images of the propagating precipitation reaction front, for the white rectangle in (a). (c) Line profiles of the intensity in images shown in (b).


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(a) C* = 0.011

(b) C* = 0.0105

(c) C* = 0.01

Figure 3. Concentration of precipitate D from the RD simulation at (a) C* = 0.011, (b) C* = 0.0105, and (c) C* = 0.01 in greyscale. The parameters for all simulations are A0 = 1.0, B0 = 0.01, DA = 1.0, DB = 0.75, and DC = 0.1.


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(a) without cysteine with cysteine

(b)

C*= 0.011 C* = 0.0105 C* = 0.01

Figure 4. Spacing coefficient as a function of band number (a) obtained from experimental results without cysteine (black circle) and with cysteine (red triangle); (b) obtained from simulation at C* = 0.011 (black), 0.0105 (blue), and 0.01 (red).


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(a)

(b)

4 mm

4 mm

1 mm

(c)

1 mm

(d)

without L-Cys

with L-Cys

Figure 5. Liesegang patterns obtained in the photochemical reduction system (a) without and (b) with cysteine. (a) Microscope image of sample without cysteine, taken at 4 h after the agarose stamp was placed on the gelatin gel. (b) Microscope image of sample with cysteine taken after 4 h. Rectangular images at the bottom: enlarged regions indicated by the white rectangle. (c) Line profiles for the sample without and with cysteine. (d) Spacing coefficient for the system with cysteine.


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TOC graphics Ligation

Non-Ligation


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Role of Nuclei in Liesegang Pattern Formation: Insights from

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