Pulse-density modulation control of chemical oscillation far from equilibrium in a droplet open-reactor system

The design, construction and control of artificial self-organized systems modelled on dynamical behaviours of living systems are important issues in biologically inspired engineering. Such systems are usually based on complex reaction dynamics far from equilibrium; therefore, the control of non-equilibrium conditions is required. Here we report a droplet open-reactor system, based on droplet fusion and fission, that achieves dynamical control over chemical fluxes into/out of the reactor for chemical reactions far from equilibrium. We mathematically reveal that the control mechanism is formulated as pulse-density modulation control of the fusion–fission timing. We produce the droplet open-reactor system using microfluidic technologies and then perform external control and autonomous feedback control over autocatalytic chemical oscillation reactions far from equilibrium. We believe that this system will be valuable for the dynamical control over self-organized phenomena far from equilibrium in chemical and biomedical studies.

. Control mechanism of fusion and fission by a droplet-fusion control program. T set j : set value of j-th fusion-fission interval; T j and w: actual interval and duration of j-th fusion-fission, respectively. p(t; T, w): pulse-train function expressing fusion-fission process. q: basal strength of chemical fluxes. First, a fusion flag in the droplet-fusion control program is turned ON by following T set j ; the droplet-fusion control program then waits for passing of a transporter to prevent unintended fusion, and AC voltage is turned ON; when the next transporter comes, the transporter fuses with the reactor; finally, just after the fission of the transporter, the flag and AC voltage are turned OFF; this cycle is repeated. The experiments were carried out in a jacketed beaker (0065-01-13-01, Tokyo Glass Kikai). The total reaction volume was 18 mL, and 75 mM KBrO 3 , 15 mM K 4 Fe(CN) 6 , 7.5 mM H 2 SO 4 , 100 mM Na 2 SO 3 , and 1 mM fluorescein sodium were flowed into the beaker at flow rates 0.9 mL min −1 using peristaltic pumps (MP-1000, EYELA). 40 • C water was run into the beaker jacket to maintain the solution temperature. Fluorescence was observed using a digital camera (EX-F1, Casio). Excitation wavelength of light-emitting diode light: 470-475 nm. Emission filter wavelength > 540 nm. The pH was monitored using a pH meter (D-52LAB, Horiba).

Supplementary Tables
Supplementary Table 1. Kinetic reaction equations and parameters of BSF chemical reactions.
No. Chemical reaction Kinetic reaction equations Kinetic parameters (ref. [3]) indicates concentrations of chemical species i in a reactor. 'fwd' and 'bwd' indicate forward and backward reactions, respectively.

Supplementary Notes
Supplementary Note 1. Details of the analytical calculation of p(t; T, w) p(t; T, w) represents a time-dependent discrete process in fusion-fission. p(t; T, w) is mathematically a square-wave function, which has two values: 0 (non-fusion state) or 1 (fusion state). T = {T j } is the fusion interval between the j-th and ( j + 1)-th fusion states (Fig. 1b). w = {w j } (w j < T j ) is the duration of the j-th fusion state. Thus, p(t; T, w) is described as where P(t, t s , t f ) represents a single fusion which starts at time t s and finishes at time t f ; H(t) is the Heaviside unit step function: H(t) = 0 (t < 0); H(t) = 1 (t ≥ 0); and τ j is the time at which j-times fusion starts (i.e. τ j+1 − τ j = T j ). p(t; T, w) expresses an arbitrary fusion-fission process as a time-dependent function. First, we consider a simple case in which the fusion-fission events are periodic, i.e. T j and w j are constant (T j = T , w j = w). Additionally, T 0 = 0 and T −l = −T l are assumed for calculation simplicity. The Heaviside unit step function, H(t), is expressed in Fourier integral form: Thus, The fusion-fission events are assumed to be periodic, τ j = jT ; therefore, where δ(ξ) is the Dirac delta function. Similarly, Thus, Because the phase difference is not essential, we have To estimate the effect of imperfect stirring of a reaction solution in the reactor, we analyze a simple two-region model, considering diffusion of chemicals between the upper and lower regions in the reactor, as follows ( Supplementary Fig. 6a): where u A = {u A i } and u B = {u B i } (i = 1, 2, · · · , n) indicate the concentrations of chemical species {U i } in the upper (A) and lower (B) regions, respectively. D is the exchange rate of chemicals resulting from diffusion between the upper and lower regions.
When a solution in the reactor is well stirred, i.e. the exchange rate is high (D = 100 min −1 ), the time courses of the pH in the upper and lower regions are approximately the same (Supplementary Fig. 6b). In contrast, when a solution in the reactor is not well stirred, i.e. the exchange rate is low (D = 1 min −1 ), the time courses of the pH in the upper and lower regions behave differently ( Supplementary Fig. 6c). If the spatial information is controlled more precisely using factors such as the droplet shape, this type of imperfect mixing may increase the system complexity [4].
Supplementary Note 4. Generation of p and q by pulse-density modulation control To generate time-variable chemical fluxes, we used where t 0 = 0; t j = j−1 l=0 T set l ; and q is the baseline value of q(t). Z q (t) was varied according to designated functions as follows: where U(−1, 1) indicates uniform random numbers between [−1, 1]; R(t, T q ) gives the residue obtained when t is divided by T q ; and S (t, T q ) gives 1 when R(t, T q ) < T q /2, and otherwise gives 0. Supplementary Figure 9 shows the generated p and q by pulse-density modulation control.
Supplementary Note 5. Fabrication of microfluidic device for droplet open-reactor system The microfluidic system was constructed using two poly(methyl methacrylate) (PMMA) plates (1 mm thickness, extruded PMMA, Mitsubishi Rayon; Supplementary Fig. 2). The microchannel was designed using three-dimensional (3D) computer aided design (CAD) software (Rhinoceros 3D, McNeel), and was fabricated on the upper plate using a fine milling machine (MDX-40A, Roland DG) with a 0.5 mm diameter endmill (MSEM230-0.5, NS Tool). The upper plate had four holes for three inlets and an outlet, and two square holes for electrodes; additionally, it had one more hole at the top of the square chamber for injection of a reactor. The upper and bottom plates were attached by ∼ 30 min of thermal compression bonding at ∼ 90 • C (AL194-L, Romanov). At the beginning of the experiments, the reactor was introduced into the square chamber through the top hole using a micropipette; the hole was then sealed with a transparent cellophane tape (Scotch 313, 3M). The oil and aqueous phases were then flowed. Details are given in Supplementary Fig. 2.
Two aqueous phases and an oil phase were introduced through inlets using ethylene tetrafluoroethylene (ETFE) tubes (Tokyo Glass Kikai) (inner diameter: 0.50 mm; outer diameter: 1.59 mm). The ETFE tubes were attached to the upper PMMA plate with an epoxide-based adhesive (AR-R30, Nichiban).