Synchronization in Autonomous Mercury Beating Heart Systems

Jun 4, 2014 - heterogeneities in the setup) MBH systems get synchronized at the ... best of our knowledge, this synchronization phenomenon involving t...
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Synchronization in Autonomous Mercury Beating Heart Systems Dinesh Kumar Verma,† Harpartap Singh,† A. Q. Contractor,‡ and P. Parmananda*,† †

Department of Physics and ‡Department of Chemistry, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India S Supporting Information *

ABSTRACT: The ability of the mercury beating heart (MBH) system to exhibit sustained mechanical and electrochemical activities simultaneously without any external agent (fluctuating or constant), has attracted researchers for decades. The interplay of these activities could mimic the biological phenomena such as a pulsating heart that occurs due to the coupled tissues exhibiting mechanical as well as electrical dynamics. In the present work, we have studied experimentally the dynamics of electrically coupled two and three autonomous MBH systems. A dynamical triangular (heart) shape, in the traditional watch glass geometry, has been chosen for the experiments. It is found that the redox potentials (electrical behavior) of the quasi-identical (due to the inherent heterogeneities in the setup) MBH systems get synchronized at the intermediate coupling strengths whereas coherence in their mechanical activities occur only at large coupling strengths. To the best of our knowledge, this synchronization phenomenon involving two distinct activities (electrical and mechanical) and different coupling thresholds has not been reported, so far. The coherent mechanical activities means the simultaneous occurrence of compressions and expansions in the coupled Hg drops, which are shown using snapshots. In addition to this, the redox time series have also been provided to demonstrate the synchronization in the electrical behavior of MBH systems. Moreover, a mathematical framework considering only electrical and mechanical components of the MBH systems is presented to validate the experimental findings that the strong synchrony in the redox potentials of the MBH systems is a prerequisite for the synchrony in their mechanical activities.

I. INTRODUCTION The behavior of synchronization wherein the interacting elements exhibit coordinated activities, has been observed in various natural phenomena, for example, the flashing of fireflies,1 the chirping of crickets,2 pacemaker cells of the heart3 and the circadian,4 and insulin secreting cells of the pancreas,5 etc. Although these biological oscillators are different in many aspects with regard to chemical composition, structures, and functions etc., they all are self-sustained oscillators. Similar to these biological elements, a mercury beating heart (MBH) system is a well-known self-sustained oscillator that has been extensively studied.6−14 The interesting feature of this system is that, in spite of being an electrochemical oscillator, the MBH system also exhibits mechanical activities along with the electrical activities. These mechanical oscillations give rise to various dynamical shapes similar to triangular (heart), elliptical, pentagonal, and hexagonal, etc., which have been reported by Castillo-Rojas et al.11 along with the underlying chemistry involved. Among all of these dynamical modes, the heartlike shape (triangular) has been found to be the most stable in the traditional watch glass geometry.12 Previously, nonautonomous MBH systems have also been reported employing a constant10 and an oscillating15 potential. Moreover, a pair of nonautonomous MBH systems shorted with a platinum wire were also investigated in constrained geometries (linear and circular).10 In contrast, we © 2014 American Chemical Society

reckon that the study of the coupled autonomous MBH systems is more compelling (which has not been reported, so far). This is because there is a need to understand how the two distinct activities (electrochemical and mechanical) exhibited by the autonomous MBH systems play the crucial roles in the emergence of their coherent behavior and to know the prerequisite for the synchrony in their mechanical behavior since it involves complex hydrodynamics. These issues have been addressed in this paper both experimentally and mathematically. In the present work, we have studied the emergence of different synchronization domains in the coupled autonomous MBH systems (in the traditional watch glass geometry exhibiting a heart shape, discussed in section II) as the magnitude of their mutual interactions is systematically varied. It has been observed that the redox potentials of these coupled MBH systems get synchronized at intermediate coupling strengths whereas the coherence in their mechanical activities emerge only at large coupling strengths (given in section III). The coherence in mechanical activities is defined as the simultaneous compressions and expansions in the coupled Hg drops. Moreover, section III also includes a mathematical Received: April 14, 2014 Revised: June 4, 2014 Published: June 4, 2014 4647

dx.doi.org/10.1021/jp503627q | J. Phys. Chem. A 2014, 118, 4647−4651

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electrodes were removed to improve the visual inspection of the Hg drops (top view in the snapshots16,17).

framework considering only electrical and mechanical components of the MBH systems to validate the experimental findings. Finally, conclusions have been made in section IV.

III. RESULTS AND ANALYSIS The autonomous dynamics of the two quasi-identical MBH systems have been presented in Figure 2 and Figure 3. The

II. EXPERIMENTAL SETUP The schematic diagram of the experimental setup for the two coupled MBH systems in the traditional watch glass geometry (similar to that reported by Anvir12 and Castillo-Rojas et al.11 for one MBH system) is presented in Figure 1. Both of the

Figure 2. Superimposed time series (a) and the state space portrait (b) of the redox potentials representing the uncorrelated autonomous behavior of two quasi-identical MBH systems.

Figure 1. (a) Schematic diagram of two coupled MBH systems. Pt wires shrouded in glass tubes are inserted into the Hg drops. Potentials of Hg drops were measured with respect to the Fe nails. The schematic diagram b is equivalent to that of a wherein Hg drops are directly connected via Pt wire to realize the R ≈ 0 Ω case.

MBH systems are designed as identical as possible such that each MBH system consists of a volume (2 mL) of mercury submerged in a pool of 0.12 mM K2Cr2O7 in 6 M H2SO4 along with an Fe nail (of 99+% purity) placed in a watch glass of 75 mm diameter and radius of curvature equal to 8.02 cm (as shown in Figure 1). Due to the inherent heterogeneities, i.e., the manufacturing error in the watch glass (≈2%) and the unavailability of the identically pointed Fe nails, the two systems are considered to be quasi-identical. To reiterate, the reason for choosing the heartlike (triangular) dynamical shape to study the synchronization phenomena in the MBH systems is its occurrence for a broad range of the concentrations of the oxidant, i.e., [0.04 mM, 0.20 mM]. Furthermore, this triangular shape persists for about 40 min without having to replenish the oxidant in the system. To follow the mechanical activities of the MBH systems, video clips were recorded and for measuring the electrical activity (redox potential) on the surface of Hg in both of the MBH systems, platinum (Pt) electrodes were inserted into the Hg drops. These Pt electrodes consist of Pt wires shrouded in glass tubes with small open tips to provide an inert and an isolated contact with the Hg drop. The redox potentials of the Hg drops were measured with respect to the Fe nail. Figure 1a shows a variable coupling resistor R incorporated between the Pt electrodes enabling a systematic variation of the coupling strength between the MBH systems. Figure 1b represents the situation wherein the MBH systems were shorted using a Pt wire to get the R ≈ 0 Ω case. Since, in this scenario, the redox potentials of both of the MBH systems (potential at the surface of Hg drops) are the same, the Pt

Figure 3. Uncorrelated mechanical activities of two uncoupled quasiidentical MBH systems using snapshots16 (top view) from a to l that are taken at an interval of 0.05 s each.

superimposed time series in Figure 2a are the redox potentials, i.e., v1 and v2 of the two systems when the key is open (Figure 1a). Figure 2b shows that there exists no correlation between the redox potentials v1 and v2 for this situation. The snapshots16 (taken at an interval of 0.05 s) of the video clip given in Figure 3, clearly indicate that the compressions and expansions occur in these uncoupled MBH systems are not correlated. Figure 4 and Figure 5 correspond to the scenario when the two quasiidentical MBH systems are coupled (key is closed in Figure 1). The value of the connecting resistor R (Figure 1) determines the magnitude of the coupling strength. Since the exchange of the electrical charges is the only mode of interactions between two MBH systems, the lower the value of R the higher will be the coupling strength. The coupling resistor R was varied systematically from 1000 Ω to 0 Ω to analyze the coupled behavior of the MBH systems. It was found that the correlation in their electrical activities (redox potentials) emerges for R ≤ 150 Ω wherein the coupled MBH systems exhibit phase 4648

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emerge only for strong couplings (R ≤ 5 Ω). This correlation in their mechanical activities, i.e., the simultaneous occurrence of expansions and contractions in the identical Hg drops, is visually perceptible only for R ≈ 0 Ω (directly connected with Pt wire) as shown in Figure 5. Now, we present a mathematical framework to validate these experimental findings. Consider the MBH systems as electromechanical oscillators whose governing equations are as follows: d2r1 dt

C1

2

C2 Figure 4. (a−c) Time series of the redox potentials of two coupled quasi-identical MBH systems; (d−f) their corresponding state space portraits for weak coupling (R = 500 Ω; no synchronization), intermediate coupling (R = 100 Ω; phase synchronization), and strong coupling (R = 5 Ω; almost complete synchronization), respectively.

dr1 dt

dv1 = f (v1) − σ(r1)v1 dt

d2r2 dt

= F(r1 , v1 , c1) − ξ

2

= F(r2 , v2 , c 2) − ξ

(1)

(2)

dr2 dt

dv2 = f (v2) − σ(r2)v2 dt

(3)

(4)

Equations 1, 2 and eqs 3, 4 represent the sets of equations for the two uncoupled MBH systems wherein r1, r2 are the average radii (macroscopic observable) of the two oscillating Hg drops (exhibiting the heart shape) from their centers and v1,v2 are the voltages on the Hg drops with respect to their Fe nails. The concentration of the oxidants (c1, c2) are considered as the parameters of these MBH systems. ξ > 0 is the dissipation coefficient and C1, C2 are the differential capacitances per unit area. Based upon the Newton’s law of motion, eq 1 and eq 3 represent the mechanical activities wherein the left-hand side (LHS) represents the accelerations of the unit masses (i.e., the effect) whereas the right-hand side (RHS) corresponds to the restoring forces (F(ri, vi, ci); i = 1, 2) and dissipative forces (ξ(dri/dt); i = 1, 2) (i.e., the cause). Equation 2 and eq 4 represent the electrical activities (same as in ref 8) wherein the LHS corresponds to the voltage fluctuations (Ci(dvi/dt), i.e., current) on the surface of the mercury drops. The first terms, f(vi), on the RHSs of eq 2 and eq 4 correspond to the exponential decay behavior whereas the second terms, σ(ri)vi, represent the short circuit situation (for more detail see ref 8). The conductance σ(ri) is nonzero only when Hg drops touch their corresponding Fe nails. Furthermore, we have assumed that the restoring forces F(ri, vi, ci) can be written as the sum of linear functions, −kri (the harmonic term), and nonlinear functions, g(vi,ci), h(ri). Here h(ri) is a positive monotonically increasing nonlinear function (i.e., for ria > rib, h(ria) > h(rib)) whereas the restoring force g(vi,ci) comes from the surface tension due to Lippmann’s electrocapillary effect6(similar to that in the work of Keizer et al.8) wherein the surface tension of the Hg drop decreases with the increase in its surface charge density and/or its surface potential (happening due to redox reactions) and vice versa. Thus, the restoring forces are F(ri,vi,ci) = −kri + h(ri) + g(vi,ci) where k is the force constant (k > ξ because in the case of Hg, cohesive forces are much stronger than the adhesive forces). It should be noted that since we are studying the coupled scenario (synchronous states) analytically not numerically, the functional forms are not required in the present work. In the coupled situation (Figure 1a), the charges on both the Hg drops get rearranged, due to which potentials on both of the Hg surfaces change. Therefore, to model the coupled situation, the terms (v2, v1/R) and (v1, v2/ R) are added in eq 2 and eq 4, respectively. To prove our claim

Figure 5. Synchronized mechanical activities of two coupled quasiidentical MBH systems using snapshots16 (top view) from a to l that are taken at an interval of 0.05 s each.

synchronization. Since the coupling increases with decreasing R, the quality of synchronization improves (means the phase difference gets reduced) when R is decreased below 150 Ω. This behavior of the coupled quasi-identical MBH systems has been illustrated in Figure 4 by plotting the time series and state space portraits of their redox potentials. It is clear from Figure 4c,f that at a strong coupling strength (R = 5 Ω) the redox potentials of the MBH systems almost follow each other. Although the synchronization in their electrical activities appears at intermediate coupling (R = 100 Ω in Figure 4b,e), the correlation between their mechanical activities starts to 4649

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MBH systems in the different coupled configurations (linear and ring) have also been studied. These coupled scenarios are modeled as

that, for the mechanical activities to be synchronized (quasiidentical MBH systems), first the electrical activities should be strongly correlated (i.e., at large coupling strength), assume (for an instance) that the mechanical activities are synchronized (r1 = r2) but the electrical activities are not. Now perturb this synchronous state (r1 = r2) with η; i.e, r1 = r2 + η. By substituting this perturbation in eq 1 and using eq 3, we obtain, (d2η/dt2) = −kη + (dh(r2)/dr1)η − ξ(dη/dt) + g(v1,c1) − g(v2,c2), where (dh(r2)/dr1) η = h(r2 + η) − h(r2) (linear approximation, since Hg drops are identical). This can be written as d2η dη +ξ + (k − λ)η = g (v1 , c1) − g (v2 , c 2) dt dt 2

d2ri dt

Ci

2

= F(ri , vi , ci) − ξ

dri dt

dvi = f (vi) − σ(ri)vi + Ii dt

(6)

(7)

Similar to eqs 1− 4, eqs 6 and 7 represent three MBH systems (i.e., i = 1, 2, and 3) whereas Ii denotes the interactions of the ith MBH system with the other two MBH systems, in the linear configuration:

(5)

⎧ v2 − v1 i=1 ⎪ R ⎪ 12 v − v2 ⎪ v1 − v2 + 3 i=2 Ii = ⎨ R 23 ⎪ R12 ⎪ v2 − v3 i=3 ⎪ ⎩ R 23

where λ = (dh(r2)/dr1). In eq 5, the dissipation coefficient ξ, is the stabilizing parameter because it tends to decay the fluctuations/errors η; i.e., it tends to bring the perturbed state back to the original state (i.e., r1 = r2). On the other hand, the parameter k′ = k − λ (say) destabilizes the synchrony since it is the restoring energy coefficient. Thus, the positive value of λ decreases the k′ (i.e., increases the stability of the mechanical synchrony) and vice versa. Moreover, λ will be positive only if the two Hg drops are identical. This is because, since their static radii (at t = 0) are equal r1(0) = r2(0), their maximum expansion and contraction limits are equal. Thus, in the synchronous state, during expansion mode (i.e., δr1a > δr1b): h(δr2a) > h(δr2b) and similarly for the contraction mode (i.e., δr1a ≤ δr1b): h(δr2a) ≤ h(δr2b), where δr1 =r1(0) ± r1(t), δr2 = r2(0) ± r2(t). To reiterate, the quasi-identical MBH systems include equal volumes of Hg drops with the same concentrations of the oxidants (i.e., c1 = c2 = c). Thus, for the scenario k′ > ξ (underdamped), it is evident that eq 5 represents the forced oscillator where, in the stable synchronous state (r1 = r2), the RHS should be zero; i.e., g(v1,c) = g(v2,c) which will be possible if v1 = v2. This will only happen when the magnitude of the coupling resistor R will be 0 Ω (Figure 1b). Hence, the simultaneous occurrence of expansions and contractions in the identical Hg drops (i.e., synchrony in their mechanical activities) is realized visually only for R ≈ 0 Ω (directly connected with Pt wire, i.e., at the maximum possible coupling strength). This has been shown in the snapshots16 of Figure 5 wherein both the Hg drops contract and expand together. These snapshots16 were taken at the same interval as was done for the case of the uncoupled systems (Figure 3). It should be noted that in Figure 5 one to one correspondence between the corresponding points on the surface of the two Hg drops might not be observed because the MBH system also exhibits small rotational behavior along with the vibrational dynamics. Therefore, only the macroscopic observables, i.e., crests (during compression modes) and troughs (during expansion modes) appearing at the centers of Hg drops, have been employed to characterize the synchrony phenomenon in the MBH systems which are evident in the snapshots.16 The snapshots shown in Figure 5, i.e., parts a, b, d, e, g, h, j, and k, correspond to the expansion modes whereas snapshots c, f, i, and l (Figure 5) represent the contraction modes. Furthermore, it has been found experimentally that the nonidentical MBH systems also show synchronous behavior (limited) in their mechanical activities when the volume mismatch in their Hg drops (where c1 = c2) is less than 40%17 (mathematical corroboration of this experimental observation is also formulated18). In addition to this, three quasi-identical

and in the ring (closed loop) configuration: v − v1 ⎧ v2 − v1 + 3 i=1 ⎪ R R13 ⎪ 12 v − v2 ⎪ v1 − v2 + 3 i=2 Ii = ⎨ R 23 ⎪ R12 ⎪ v1 − v3 v − v3 + 2 i=3 ⎪ R 23 ⎩ R13

where R12, R13, and R23 are the resistances of the three resistors used to connect the three MBH systems. Although, in the case of ring configuration, the synchrony in the redox potentials emerges earlier than in the linear configuration (i.e., synchrony observed at the higher values of R12 and R23), the synchrony in the mechanical activities was observed only at R12 ≈ 0, R13 ≈ 0, and R23 ≈ 0 for both of the cases. Similar to Figure 5, Figure 619

Figure 6. Synchronized mechanical activities of three linearly coupled quasi-identical MBH systems using snapshots (top view) from a to h that are taken at an interval of 0.05 s each. 4650

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(2) Strogatz, S. H.; Stewart, I. Sci. Am. 1993, 269, 102−109. (3) Jalife, J. J. Physiol. 1984, 356, 221−243. (4) Winfree, A. T. J. Theor. Biol. 1967, 16, 15−42. (5) Sherman, A.; Rinzel, J.; Keizer, J. Biophys. J. 1988, 54, 411−425. (6) Lippmann, G. Ann. Phys. 2S 1873, 149, 546−561. (7) Hoff, H. E.; Geddes, L. A.; Valentinuzzi, M. E.; Powell, T. Cardiovasc. Res. Cent. Bull. 1971, 9, 117−130. (8) Keizer, J.; Rock, P. A.; Lin, S. W. J. Am. Chem. Soc. 1979, 101, 5637−5649. (9) Lin, S. W.; Keizer, J.; Rock, P. A.; Stenschke, H. Proc. Natl. Acad. Sci. U. S. A. 1974, 71, 4477−4481. (10) Smolin, S.; Imbihl, R. J. J. Phys. Chem. 1996, 100, 19055−19058. (11) Castillo-Rojas, S.; Gonzalez-Chavez, J. L.; Vicente, L.; Burillo, G. J. Phys. Chem. A 2001, 105, 8038−8045. (12) Avnir, D. J. Chem. Educ. 1989, 66, 211−212. (13) Olson, J.; Ursenbach, C.; Birss, V. I.; Laidlaw, W. G. J. Phys. Chem. 1989, 93, 8258−8263. (14) Kim, C. W.; Yeo, I.-H.; Paik, W.-K. Electrochim. Acta 1996, 41, 2829−2836. (15) Verma, D. K.; Contractor, A.; Parmananda, P. J. Phys. Chem. A 2013, 117, 267−274. (16) Video clip (mov1.avi) of the mechanical activities of the quasiidentical MBH systems is provided in the Supporting Information. (17) Video clip (mov2.avi) of the mechanical activities of the heterogeneous MBH systems (volumewise) is provided in the Supporting Information. (18) In the case of nonidentical MBH systems, with the increment of volume mismatch in the Hg drops leads to h(r2 + η) − h(r2) = (dh(r2/ dr1)η + 0.5(d2h(r2)/dr12)η2 + (higher order terms of η depending on the mismatch) that also increases k′ as (dh(r2)/dr1) < 0. But since the absence of an energy source term (i.e., the counter to the dissipation term −ξ(dη/dt)) in eq 5, error/ fluctuations η cannot be sustained at v1 = v2. (19) Video clip (mov3.avi) of the mechanical activities of the quasiidentical three MBH systems is provided in the Supporting Information. In addition, video clip named mov4.avi showing the coupled behavior of three MBH systems in a closed loop (ring) configuration has also been given.

represents the coordinated mechanical activities of three coupled quasi-identical MBH systems (using two platinum wires, i.e., R12 ≈ R13 ≈ R23 ≈ 0) in a linear configuration, i.e., no flux boundary condition. Moreover, it has been found that closed loop (ring) coupling configuration (using three platinum wires) also yields similar results because the magnitude of the interactions remains same (as R12 ≈ R13 ≈ R23 ≈ 0) in both of these cases. For the stability analysis of the coherent behavior in the case of three MBH systems, two MBH systems, i.e., a pair, have to be considered (out of three) at a time. This results in eq 5 being like the evolutionary equation of the perturbed state (error) for a pair of MBH systems. Hence we get two evolutionary error equations (for instance in η12 and η23) in the case of linear configuration and three evolutionary error equations (for instance in η12, η23, and η31) in the case of ring configuration. Furthermore, for both of these scenarios, it has been found that the strong synchrony in the redox behavior of the three MBH systems is the prerequisite for the synchrony to be emerged in their mechanical activities (similar to the case of two MBH systems).

IV. CONCLUSION The present work reports the emergence of synchronization phenomenon in the coupled (i.e., via resistor) MBH systems in a traditional watch glass geometry exhibiting heart shape (triangular) dynamics. It has been concluded that, for the quasiidentical MBH systems, the synchrony in their redox potentials is a prerequisite for the synchronization in their mechanical activities. Moreover, the limitation of getting practically 0 Ω resistance and inherent heterogeneity (due to the parameters such as dimensions of the watch glass) in the MBH systems precludes the complete synchronization in their mechanical activities. To reiterate, a mathematical framework considering only electrical and mechanical components of the MBH systems is presented to validate the experimental findings that the strong synchrony in the redox potentials of the MBH systems is a prerequisite for the synchrony in their mechanical activities.

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ASSOCIATED CONTENT

* Supporting Information S

Four video clips showing the mechanical activities of the MBH systems (coupled as well as uncoupled) for (i, mov1.avi) two quasi-identical MBH systems, (ii, mov2.avi) two heterogeneous (volumewise) MBH systems, (iii, mov3.avi) three quasiidentical MBH systems coupled in a linear configuration, (iv, mov4.avi) three quasi-identical MBH systems coupled in a closed loop (ring) configuration. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from DST (India) and IIT Bombay is acknowledged. REFERENCES

(1) Hanson, F. E. Fed. Proc. 1978, 37, 2158−2164. 4651

dx.doi.org/10.1021/jp503627q | J. Phys. Chem. A 2014, 118, 4647−4651