Synchronization of Two Self-Oscillating Gels Based on Chemo

Article pubs.acs.org/JPCB

Synchronization of Two Self-Oscillating Gels Based on ChemoMechanical Coupling Kentaro Ito,† Takato Ezaki,† Shogo Suzuki,† Ryo Kobayashi,† Yusuke Hara,‡ and Satoshi Nakata*,† †

Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526, Japan Research Institute for Sustainable Chemistry, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba Central 5, 1-1-1 Higashi, Tsukuba 305-8565, Japan

‡

ABSTRACT: Two coupled polymer gels, showing volume oscillation caused by the Belousov−Zhabotinsky (BZ) reaction, were investigated to understand the system composed of mechanically coupled chemical oscillators. The two gels were connected with a movable plastic sheet in between and placed under constant compression. Synchronization between two identical gels occurred in a range of compression ratios. The phase difference between the two oscillating gels was not zero; instead, they showed alternate swelling−deswelling oscillations. Similar phenomena were also observed with gels of different sizes and natural oscillation periods. The experimental results suggest that a physical change in one gel can lead to a chemical change in the other and vice versa. These results were qualitatively reproduced by a mathematical model based on coupled chemical oscillators.

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INTRODUCTION Rhythmic phenomena in living organisms are important to keep them alive.1 When two or more oscillators are coupled, the phase difference between them is often locked at a constant value.2,3 Such a phenomenon is called “synchronization”, which is characteristic of rhythm observed in nonlinear systems.1 Synchronization has been well observed in living organisms, for example, plasma streaming in slime molds,4,5 action potential of cardiac cells,6 circadian rhythm,1 and locomotion;7,8 however, synchronization in living organisms is complicated due to the interplay of many factors. Therefore, simple nonliving systems and mathematical models are important not only for determining the intrinsic factors of synchronization but also for clarifying the mechanism involved. In this study, we focused on synchronization based on chemo-mechanical coupling. The Belousov−Zhabotinsky (BZ) reaction has been well studied as a typical self-oscillating chemical reaction.9,10 The catalyst oscillates between the oxidized and reduced states during the BZ reaction. A self-oscillating polymer gel showing swelling and deswelling can be realized by introducing the BZ reaction into the gel.11−14 Such a self-oscillating gel can be used as a novel actuator that mimics chemo-mechanical transduction processes in muscle that lead to locomotion. Recently, it has been reported that the period of swelling and deswelling oscillation of the BZ gel can be entrained by using a periodic force as a forced oscillator.15 Features of synchronization between two or four self-oscillating gels can be changed depending on the distance between them.16,17 These results suggest that change in the gel volume caused by the external periodic force gives feedback to the BZ reaction. © XXXX American Chemical Society

In this study, we investigate the coupling of two BZ selfoscillating gels under constant compression. Synchronization with a phase difference between the two oscillators was observed. Experimental results were qualitatively reproduced by a mathematical model based on the mechanically coupled chemical oscillators.

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EXPERIMENTAL SECTION An N-isopropyacrylamide gel (NIPAAm) was synthesized based on a procedure reported by Yoshida et al.15,18 Preparation of poly(NIPAAm-co-Ru(bpy)3-co-AMPS) gel used in the present study was carried out as follows. NIPAAm (623.7 mg), Ru(bpy) 2(vmbpy) (16.2 mg), N,N′-methylenebis(acrylamide) (MBAAm, 11.2 mg, cross-linker), and 2,2′azobis(isobutyronitrile) (AIBN, 26.6 mg, initiator) were dissolved in a mixture of methanol (2 mL) and dimethyl sulfoxide (DMSO, 0.4 mL). 2-Acrylamido-2-methylpropanesulfonic acid (AMPS, 22.0 mg) was dissolved in pure water (1.6 mL). The two solutions were mixed together as a monomer solution, and O2 in the monomer solution was removed by N2 purging for 15 min. The monomer solution was then injected into the space between two glass plates (Matsunami Glass, 76 mm × 52 mm × 1.5 mm) gapped with a silicone sheet (thickness: 1.5 mm). The injected solution was then polymerized at 60 °C for 18 h to obtain the poly(NIPAAmco-Ru(bpy)3-co-AMPS) gel. The gel was soaked in pure ethanol (50 mL) for 1 week and then soaked in a series of ethanol/ Received: January 27, 2016

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DOI: 10.1021/acs.jpcb.6b00873 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B water solutions (75, 50, 25, and 0 vol %) for 2 days each to remove unreacted monomers and to hydrate the gel. In the experiment, two pieces of the gel (2 mm × 2 mm × 1.5 mm, 2 mm × 2 mm × 1 mm, or 2 mm × 2 mm × 2 mm) were sandwiched between an acrylic square bar (Acrylite, Mitsubishi Rayon) and the wall of the glass vessel containing a BZ solution (4 mL) composed of 0.084 M NaBrO3, 0.0625 M malonic acid, and 0.894 M nitric acid.19−21 The gel was fixed or compressed by moving the acrylic bar with a stepping motor (COMS, PM80B-200 K, Japan, minimum migration length: 14 μm), as shown in Figure 1. We define the gel closer to the

Figure 1. (a) Schematic illustration of the experimental system on two BZ gels separated with a polyester sheet. They were fixed or compressed by moving the acrylic bar with a stepping motor under a constant compression ratio (CR). (b) Magnified illustration of part of (a) (side view). (c) Example of snapshot of gels (bottom view).

Figure 2. Experimental results regarding (a) spatiotemporal patterns (upper) and time-course of green level for gels 1 (solid line) and 2 (dotted line) (lower) and distance from the acrylic square bar to the polyester sheet (gray line). (b) Return map for the oscillation period on panel (a). The experimental conditions are (1) direct contact without a plastic sheet, CR ≈ 0%; (2) indirect contact with the plastic sheet, CR ≈ 0%; and (3) indirect contact, CR = 30%.

acrylic bar as gel 1 and the other one further away as gel 2. A polyester sheet (5 mm × 5 mm × 0.1 mm) was placed between the gels to separate them from each other. The compression ratio (CR) was defined as (1 − (Δh/h0)) × 100% (where Δh is thickness of gel during compression and h0 us thickness of gel at the reduced state without compression). The minimum distance between the acrylic bar and the wall of the glass vessel, Δh, was constant under compression. The compressed pressure ranged between 0 and 40 kPa. In this experiment, oscillation of the gel was maintained for at least 5 h. The experiments were performed in a thermostat water bath (Yamato Scientific, CLH300, Japan) at 18.0 ± 0.5 °C.15,18,19,21 The oscillations of the gels were monitored with a digital camera (Canon, EOS kiss Digital N, Japan; time interval: 10 s) from below the water bath (see Figure 1). Snapshot images of the gel were analyzed by an image processing system (National Institutes of Health, ImageJ). The green level was extracted from the RGB image to evaluate the oscillatory phenomena of the gels15 because green and orange colors corresponded to the oxidized and reduced states of the gel, respectively. In each gel, the chemical oscillation of the BZ reaction was almost in-phase (i.e., with no phase lag), with the swelling and deswelling mechanical oscillation of the gel.20

period, T0 and Ta are the period for gel 1 and the time interval between the peak in gel 1 and the successive peak in gel 2, respectively (illustrated in Figure 2a-2). We define synchronization as when points on the return map are concentrated within 10% of the map area, as shown in Figure 2a-3. In contrast, when their distribution is over 10% area, especially around the diagonal line with Ta(n)/T0(n) = Ta(n + 1)/T0(n + 1), we define the behavior as nonsynchronization, as shown in Figure 2a-2. When two gels were in direct contact with each other without a plastic sheet, in-phase synchronization was observed, as shown in Figure 2a-1. That is, gels 1 and 2 oscillated in step with each other, and the points on the return map clustered tightly around at the origin (0, 0) in Figure 2b-1. In contrast, when the two gels were divided with a plastic sheet at CR ≈ 0%, they oscillated independently, as shown in Figure 2a-2. The phase difference between two gels progressively changed with time, and the points of the return map in Figure 2b-2 lay near the diagonal line, Ta(n)/T0(n) = Ta(n + 1)/T0(n + 1); that is, synchronization was not observed. The swelling and deswelling of the two gels were characterized by the fluctuation of d (the minimum distance between the acrylic bar and the plastic sheet), as seen in Figure 2a-2. During one cycle of oscillation, d attained the maximum value when gel 1 swelled and the minimum when gel 2 swelled. When the two gels were compressed, they appeared to be synchronized but never oscillated in-phase. Instead, the points on the return map were clustered near one point in the range of 0.1 to 0.9 on the diagonal line, as shown in Figure 2b-3. We defined such behavior as “out-of-phase synchronization”.

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RESULTS First, we examined the oscillations under three experimental conditions: (1) direct contact of two equivalent gels (2 mm × 2 mm × 1.5 mm) without a plastic sheet under no compression (CR ≈ 0%), (2) indirect contact with a plastic sheet in between and CR ≈ 0%, and (3) indirect contact with a plastic sheet in between and under compression (CR = 30%). Figure 2 shows (a) the space-time diagrams of color change in the gels and (b) the corresponding return maps. The return map (Ta(n)/T0(n) versus Ta(n + 1)/T0(n + 1)) is used to evaluate the phase difference between the oscillating gels; for the nth oscillation B


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the oscillation periods for the respective gels. In this experiment, gel S was closer to the acrylic bar and gel L was further away with the plastic sheet in between. As shown in Figure 4a, the periods for both gels increased with an increase in CR, and they were very close to each other at ∼30% but were away from that for gel L over 30% again. This is also shown in Figure 4b, where TS/TL = 1 at around CR = 30%; that is, the gels were synchronized. The time course at CR = 26% is plotted in Figure 4c, showing that the synchronization is out-of-phase. On the contrary, TS/TL > 1 when CR was smaller than 25% or larger than 35%; that is, synchronization was not observed.

We define the ratio of out-of-phase synchronization as (Nout/ Nall) × 100 (where Nall is number of experimental runs and Nout is number of out-of-phase synchronization). Figure 3 shows the

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DISCUSSION On the basis of the experimental results and related works,15−17,20,22,23 we discuss the relationship among the BZ reaction, compression ratio, and swelling−deswelling oscillation of the gels. Figures 2 and 3 suggest that gels coupled via the plastic sheet showed out-of-phase synchronization under a range of compression ratios. No in-phase synchronization was observed for such coupled gels because the chemicals (activator and inhibitor) that induce the BZ reaction could not be exchanged between the gels. Figure 4 suggests that 1:1 synchronization can be generated under certain values of the compression ratio when two gels of different sizes are coupled. It has been reported that the oscillation period of gel decreases with an increase in the size of the gel because the decrease rate of the activator concentration in an ion-exchange bead is proportional to area-to-volume ratio of the bead.22,23 In this experiment, the oscillation period of the small gel was found to be 10% longer than that of the large gel. It has also been reported that the oscillation period increases with an increase in the compression ratio.15 The two gels with different frequencies cannot oscillate inphase for the following reason. Were they oscillating in phase, the plastic sheet would not move, as it would always experience two equal and opposite forces. This is equivalent to the case where each gel is placed between two fixed walls individually. Thus, they should oscillate with their own natural frequencies, contradicting our initial assumption that they oscillate in-phase; however, they can interact through the movement of the plastic sheet with a phase difference, and the frequencies of oscillations are modified by this effect. It seems that the same number of the phase modifications are applied to gels because the lengths of strokes are the same throughout the period; however, the same number of phase modifications will never lead to synchronization of gels that have different natural frequencies. Thus, we can conclude that the different amounts of phase modifications are applied despite the sharing same plastic sheet boundary. Figure 5 schematically shows the mechanical process of such out-of-phase synchronization. They are both in the reduced state in the first stage (State I). After gel 1 oxides (green) and swells, it compresses gel 2 (State II). Gel 1 reaches a maximum thickness and then shrinks. Now, gel 2 oxidizes and starts to swell just after gel 1 stops swelling (State III); however, its volume has not changed greatly, due to the pressure from the already swelled gel 1. Then, gel 2 swells accompanying the shrinking of gel 1 (State IV). Finally, gel 2 shrinks and the system returns to State I. It should be noted that the two gels oscillate with the same periods, but the phases in their swelling−deswelling oscillations are different. From this observation, we made a hypothesis that the modification of the phase velocities, which is induced by volume change, depends on not only the amount of volume change but also the phase of

Figure 3. Experimental results regarding the relationship between the compression ratio (CR) and the ratio of out-of-phase synchronization for the two equivalent gels. The total number of examination was 50. Out-of-phase synchronization was observed at CR = 15−50%. Gels were broken at CR > 60%. The experimental conditions correspond to those of Figure 2a-2,a-3.

relationship between this ratio of out-of-phase synchronization and the compression ratio (CR) when the two equivalent gels were in indirect contact with the sheet. The two gels were broken when CR was higher than 60%, while no synchronization was observed when CR was 55%. Out-of-phase synchronization was observed in the range of 20 ≤ CR ≤ 50%, and the ratio of out-of-phase synchronization was the highest at CR = 35−40%. Next, we examined two gels of different sizes to clarify the coupling between two oscillators with different oscillation periods. When not compressed, the oscillation period for the smaller gel (S, 2 mm × 2 mm × 1 mm) was longer than that for the larger one (L, 2 mm × 2 mm × 2 mm): TS = 5.5 min and TL = 5.0 min. Figure 4 shows the relationship between / and CR, where and are the average values of

Figure 4. Experimental results regarding the ratio of the oscillation periods of different sized gels (S: 2 mm × 2 mm × 1 mm, L: 2 mm × 2 mm × 2 mm). (a) Average periods of gel S (, filled triangle) and gel L (, empty square), (b) / (empty circle) as a function of the compression ratio, CR, and (c) time-course of green level for gel S (solid line) and gel L (dotted line) at CR = 26%. C


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Figure 5. Schematic illustration of the mechanism of out-of-phase synchronization. The reaction rate of the physically compressed gel is suppressed when the other gel swells, and physically compressed gel is back to the general reaction rate when the other gel deswells.

Figure 6. Schematic illustration showing comparison between experimental and numerical models.

the moment at the given time. We will discuss a mathematical model of the physically coupled gels to check the hypothesis. Let us consider a model of the two mechanically coupled BZ gels. Figure 6 is a schematic illustration of our mathematical model. To describe the relationship between the applied force and deformation on the gel, we consider that the gel is composed of a spring and a damper. The natural length of the spring, li (i = 1 or 2), varies periodically depending on the oxidation−reduction state of the gel. The actual length, hi (i = 1 or 2), follows up the natural length. The damper plays a role in the effect of gradual follow-up. It should be noted that unlike a pure mechanical model, the period of gel oscillation is not governed by its spring coefficient and damper resistance but by the period of the chemical reaction. The thicknesses of gels 1 and 2 are described by h1 and h2, respectively; h1 + h2 = H, where H is the distance between the acrylic bar and the wall. We assume that each gel has a linear elasticity and a variable natural length, li (i = 1, 2). li depends on the phase variable ϕi that describes the oxidation−reduction state of the gel. The natural length is given by the following equation.

li = Li(1 + a cos ϕi)

(1)

where Li > 0 and a = 0.05. When cos ϕi > 0 and cos ϕi < 0, gel i is in its oxidized and reduced states, respectively. The relationship between thickness hi and the compressive force Fi can be described as follows by Hooke’s law

Fi = κ(li − hi)/li

(2)

where κ is the stiffness of the material and κ/li acts as a spring coefficient if li is constant. Neglecting inertia, the relationship among the spring forces and the damping forces leads to the following equation

ηv = AF

Figure 7. (a) Numerical results regarding (a) phase dependence of the intrinsic phase velocity, f ̃ (rL , L , ϕ). f ̃ (rL , L , ϕ) has the minimum value fc̃ (rL , L) at ϕ = π − Φf . As r becomes smaller, fc̃ (rL , L) is reduced, leading to a longer period of oscillation, (b) compression ratio dependence of fc̃ (rL , L), and (c) the ratio of fc (rL1̃ , L1) to fc (rL̃ 2 , L 2) at κ /̃ η ̃ = 3.4.

(3)

where η is the coefficient of viscosity, v = ⎛ F1 ⎞ v1 d ⎛ h1 ⎞ 1 1 −1 v2 = dt ⎜⎝ h ⎟⎠ , F = ⎜⎝ F2 ⎟⎠, and A = 2 −1 1 . The derivation 2 of eq 3 is shown in Appendix I.

( )

(

)

D


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where α is a positive coupling constant. f(hi, Li, ϕi) is the intrinsic phase velocity and given by the following f (hi , Li , ϕi) = ω(tanh(q − p) + tanh(p + q cos(ϕi + Φf )) + fc (hi , Li)) ⎛h ⎞ 2β exp( −γ(1 − hi /Li)) f (hi , Li) = λ⎜ i − − ξ⎟ hi ⎝ Li ⎠

(5)

(6)

where p = 7, q = 10, Φf = π/4, and Φg = −π/3. It should be noted that eqs 5 and 6 are designed to qualitatively reproduce the period elongation by increasing compression rate or reducing the original size of the gels. While the phase velocity is usually a constant (e.g., in a harmonic oscillator) in our model, dϕi/dt depends on ϕi because it allows a simple description of relaxation oscillation by using this form. As fc (hi, Li) becomes smaller, the period of oscillation becomes longer. The first term in the bracket of eq 6 denotes the effect of the concentration of catalyst, and the second term in the bracket on the right side denotes the effect of the surface in contact with glass or plastic sheet on the surface area-to-volume ratio of the gels (see Appendix II). The last term in the right-hand side of eq 4 is based on our hypothesis that an increase in the gel volume may influence the chemical reaction. We assume that only inflow of reactants influences reaction speed and the intensity of the effect depends on the relative volume change. Thus, we put g(hi, vi) = max{vi/hi, 0}. We also assume that the influence of chemical reaction speed depends on the chemical state ϕi and set cos(ϕi + Φg) as the phase response. Therefore, g(hi, vi) is multiplied by cos(ϕi + Φg) in eq 4. Before starting the numerical simulation, we recast the model equations in dimensionless variables. We take the thickness of the nonpressed gel as the characteristic length h0 = 1.5 mm, and the gel oscillation period at CR ≈ 30% as the characteristic time τ = 8 min. The dimensionless thickness and time are hĩ = hi/h0 and t ̃ = t/τ, respectively. Thus, the model equations becomes

Figure 8. Numerical results regarding the ratio of the oscillation period of two gels with different sizes (L1̃ = 4/3, L̃ 2 = 2/3). (a) Average periods of gels 1 and 2 at α = 46.0. (b) / at α = 45.0, 46.0, and 47.0. A region of synchronization appears at α = 46.0 and 47.0. These results show the same tendency of the experimental result shown in Figure 4.

Fi ̃ =

κ̃ ̃ (li − hĩ ) li ̃

(7)

dϕi /dt ̃ = f ̃ (hĩ , Lĩ , ϕi) + αg (hĩ , vĩ) cos(ϕi + Φg )

(8)

f ̃ (hĩ , Lĩ , ϕi) = ω̃ (tanh(q − p) + tanh(p + q cos(ϕi + Φf )) + fc̃ (hĩ , Lĩ ))

Figure 9. Numerical results regarding time course of (a) cos ϕ1 and cos ϕ2, and (b)h̃1 at CR = 30%. The positive and negative values of the vertical axis in panel (a) correspond to the oxidized (Ox) and reduced states (Red), respectively. The parameters used in the numerical simulations were ω̃ = 9.5, α = 46.0, and H̃ = 1.4.

⎞ ⎛ h̃ 2β exp( −γ(1 − hĩ /Lĩ )) fc̃ (hĩ , Lĩ ) = λ⎜ i − − ξ⎟ ̃ ̃ hi ⎠ ⎝ Li

(10)

η̃ν̃ = AF ̃

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⎛ h̃ ⎞ ⎛ F̃ ⎞ ⎛ ν1̃ ⎞ 1 1 d ν ̃ = ⎜ ⎟ = dt ̃ ⎜⎜ ⎟⎟, F ̃ = ⎜⎜ ⎟⎟, ⎝ ν2̃ ⎠ ⎝ F2̃ ⎠ ⎝ h2̃ ⎠ κ /̃ η ̃ = τκ /h0η, λ = 0.3, γ = 3.3, β = 0.07, ξ = 0.09, and H̃ = H /h0 . To check the model prediction against our experimental 2 4 results, we set L1̃ = 3 and L̃ 2 = 3 , which correspond to our second experiment with differently sized gels. Before starting the numerical simulation, we briefly examine the behavior of f ̃ (h ̃, L̃ , ϕi). As r( = h /̃ L̃) becomes larger, the oscillation where

The phase oscillators ϕ1 and ϕ2 are used to describe the relaxation of oscillation of the BZ gels. It has been reported that compression leads to an increase in the gel oscillation period,3 and the period also increase as the size of gel decreases.22,23 To mimic this property, we give the dynamics of phases ϕ1 and ϕ2 by the following equation. dϕ/dt = f (hi , Li , ϕi) − αg (hi , vi) cos(ϕi + Φg )

(9)

(4) E

ω̃ = τω = 9.5,


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it into eq 12, we eliminate the term Fc and the equation can be rewritten as

period of each gel becomes shorter, as shown in Figure 7a,b. On the contrary, Figure 7c shows that the ratio fc̃ (rL̃ , 4/3)/fc (rL̃ , 2/3) is not a monotonic function of r, which means the gels can easily to synchronize in a range of r values, that is, compression ratios. Figure 8 shows the numerical results of the synchronized motion under compression. The compression rate CR is defined as (1 − H̃ /(L1̃ + L̃ 2)) × 100%. There is a small region around CR = 30% in which the two gels have the same period of oscillation, as shown Figure 8a. Figure 8b shows the relationship between / and CR. Here and are the average values of the oscillation periods for gels 1 and 2 in the successive time evolution, respectively. The parameter region for synchronization, that is, / = 1, corresponds well to the experimental result shown in Figure 4b at α = 46.0. The time course of cos ϕi in the region of synchronization is shown in Figure 9. The peaks in cos ϕi coincide with the green level of gel i in our model. The out-of-phase synchronization is apparent from the phase difference in Figure 9a.

m

(−11

)

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APPENDIX II: DERIVATION OF THE MODEL EQUATION, EQUATION 6 In our model, fc(hi, Li) has a role in determining the period of oscillation of gel i. Several factors, for example, the catalyst concentration and surface area-to-volume ratio, are known to affect the period of oscillation. Assuming that top and bottom surface area (1.5 mm × 1.5 mm) of the gel do not change with compression, the volume per catalyst molecule in gel is proportion to hi/Li. In this model, we define fc(hi, Li) as the sum of their effects per the following equation. ⎛h ⎞ βS fc (hi , Li) = λ⎜ i − − p⎟ V ⎝ Li ⎠

CONCLUSIONS We investigated the oscillation of two coupled gels in the BZ reaction. Out-of-phase synchronization was observed when the gels with different natural periods were connected with a sheet in between and within a range of compression ratio. Out-ofphase synchronization was also generated when the oscillation periods of two gels with different sizes were different. The mechanism of out-of-phase synchronization was discussed in relation to the alternating between the compression and expansion of the gels induced by chemo-mechanical interaction. The present study suggests that chemo-mechanical coupling can be used for swelling and deswelling synchronization. A phenomenological model based on the coupled chemical oscillator model qualitatively reproduces the observed out-ofphase synchronization. It should be noted that our model includes several assumptions; however, from the viewpoint of the collective behavior of actuators, BZ gels only have physical coupling and no signal coupling or the force sensors, and our model can describe such a case without resorting to the detailed chemical components.

(14)

where S is the surface area of gel in contact with the solvent and V = hid2 is the volume of gel. λ, β, and p are constant values. In the experiment, the top and bottom surfaces are also in contact with the solvent. Here we assume that their contribution is exponentially decreased by hi/Li. Thus, we define the contacting area of surface as ⎛ ⎛ h ⎞⎞ S = 4hid + 2d 2 exp⎜⎜ −γ ⎜1 − i ⎟⎟⎟ Li ⎠⎠ ⎝ ⎝

where γ is a positive constant. Substituting S into eq 14 gives eq 6 where ξ = p + 4β/d.

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AUTHOR INFORMATION

Corresponding Author

*Tel/Fax: +81-82-424-7409. E-mail: [email protected]. jp. Notes

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The authors declare no competing financial interest.

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APPENDIX I: DERIVATION OF THE MODEL EQUATION, EQUATION 3 To establish the model equation (eq 3), we considered the following two cases. In the first case, two gels in contact push against each other with force F0 > 0, and h1 + h2 = H holds. In the second case, gels are not in contact and oscillate independently. We consider only the first case because the second one is only approximation under small compression ratio, that is, CR < a × 100%. dh The rate of change in thickness vi = dti (i = 1, 2) obeys the following equation of motion dv + ηv = F − Fc dt

1 2

(13)

−1 . F = (F + F )/2 can be easily 0 1 2 1 obtained from these equations. The term m (dv/dt) in eq 13 can be eliminated by letting m → 0; thereby eq 3 is obtained.

where A =

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m

dv + ηv = AF dt

ACKNOWLEDGMENTS We thank Professors Hiroyuki Kitahata (Chiba University, Japan) and Nobuhiko J. Suematsu (Meiji University, Japan) for their helpful discussion. This study was supported in part by a Giant-in-Aid for Scientific Research 25410094 for S.N., Grantin-Aid for Young Scientists (B) 24740066 for K.I, and JSPS Bilateral Joint Research Project between Japan and the Polish Academy of Sciences.

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REFERENCES

(1) Winfree, A. T. The Geometry of Biological Time; Springer: New York, 1980. (2) Pikovsky, A.; Rosenblum, M.; Kurths, J. Synchronization: A Universal Concept on Nonlinear Sciences; Cambridge University Press: Cambridge, U.K., 2001. (3) Kuramoto, Y. Chemical Oscillations, Waves, and Turbulence; Springer: New York, 1984. (4) Nakagaki, T.; Yamada, H.; Ueda, T. Interaction Between Cell Shape and Contraction Pattern in the Physarum Plasmodium. Biophys. Chem. 2000, 84, 195−204.

(12)

⎛F ⎞ v where m is mass of polyester sheet, v = v21 , F = ⎜ F1 ⎟, and ⎝ 2⎠ ⎛ F0 ⎞ Fc = ⎜ F ⎟ represents the contact force between two gels. The ⎝ 0⎠ constraint condition h1 + h2 = H requires v2 = −v1. Substituting

( )

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Synchronization of Two Self-Oscillating Gels Based on Chemo

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