Artificial temperature-compensated biological clock using temperature-sensitive Belousov–Zhabotinsky gels

The circadian rhythm is a fundamental physiological function for a wide range of organisms. The molecular machinery for generating rhythms has been elucidated over the last few decades. Nevertheless, the mechanism for temperature compensation of the oscillation period, which is a prominent property of the circadian rhythm, is still controversial. In this study, we propose a new mechanism through a chemically synthetic approach (i.e., we realized temperature compensation by the Belousov–Zhabotinsky (BZ) gels). The BZ gels are prepared by embedding a metal catalyst of the BZ reaction into the gel polymer. We made the body of BZ gels using a temperature-sensitive polymer gel, which enabled temperature compensation of the oscillation by using temperature dependence of volume. Moreover, we constructed a simple mathematical model for the BZ oscillation in temperature-sensitive gels. The model can reproduce temperature compensation of BZ gels, even though all reactions are temperature sensitive according to the Arrhenius rule. Our finding hints that a soft body coupling may be underlying temperature-compensated biological functions, including circadian rhythms.


Synthesis of BZ gels
For the PNIPAAm BZ gels, we prepared the temperature-sensitive BZ gels in the following way: We added 0.573 g of NIPAAm, 0.00813 g of MBAA as a crosslinker, 0.0712 g of ruthenium(4-vinyl-4-methyl-2,2-bipylridine) bis (2,2-bipyridine) bis (hexafluorophosphate) [Ru(bpy) 3 ] 2+ ) and 0.00866 g of AIBN as an initiator to methanol (1.5 ml). We then stirred the resulting solution for 30 min. We added 0.0182 g of 2acrylamido-2-methylpropanesulfonic acid (AMPS) to pure water (1.5 ml) and stirred the resulting solution. We mixed two solutions together. We then stirred the mixed solution and purged it with dry nitrogen gas. We injected the mixed solution into glass capillaries with a diameter of 1.0 mm. We put these glass capillaries into an oven at 60 ºC for 20 h.
We finally washed the gel samples carefully. The reaction solution of the BZ reaction was prepared by mixing nitric acid, sodium bromate and malonic acid in the ratio of 1:1:3 in the Molar concentration.
For the PAAm BZ gels, which are non-temperature-sensitive, we followed the same procedure except for the quantity of chemicals as following way: We added 0.528 g of AAm, 0.00119 g of MBAA, 0.104 g of [Ru(bpy) 3 ] 2+ and 0.00866 g of V-50 as an initiator to methanol (1.5 ml). We then stirred the resulting solution for 30 min. We added 0.0266 g of AMPS to pure water (1.5 ml) and stirred the resulting solution. The subsequent synthesis process for the PAAm BZ gels is similar to that described above.

Observation and data analysis
For the observation, the BZ gels are placed in a constant temperature bath filled with the reaction solution (Supplemental Figure S1). The temperature is controlled by circulating water. We recorded videos of the BZ gels from vertical direction. Supplemental Videos are the examples. S1, S2, S3 and S4 correspond to the cases for PAAm BZ gel at 15℃, PAAm BZ gel at 40℃, PNIPAAm BZ gel at 15℃ and PNIPAAm BZ gel at 30℃, respectively. All of them are 256 times speed. To obtain the time series of hue value, we fixed a pixel of the BZ gel in the video and calculated the hue (Fig. 3).
Fig. S1 Schematics of the experimental system.

Estimation of period
Here, we estimate the oscillation period of the limit cycle. Supplemental Figure S2 shows a numerically obtained trajectory of the limit cycle and nullclines of differential equations (8) and (9) in the main text. Purple points show the trajectory of limit cycle at a certain interval. The trajectory moves counterclockwise as increases and it stays most of the time in a narrow domain between the cubic nullcline and the axis. So, the period of the limit cycle is estimated by the time for the trajectory to pass the domain.
The nullclines of differential equations (8) and (9) Now we denote the range of trajectory as and the time for the state to pass as Δ .
Then, denoting the maximum and minimum of in as max and min , respectively, max and min can be estimated approximately as follows. As shown in Fig. S1, max and min are almost the same as the maximal and minimal value of the nullcline (Fig.   S2). Near the maximal point, since ≫ (1 − ) 2 , Eq. (S1) is approximated as Then, we obtain The minimal point of nullcline (S1) emerges at ≃ (1 − ) 2 . Thus, we can estimate the minimal value by substituting it into Eq. (S5), which becomes and Eq. (S2), respectively. Purple dots show limit cycle plotted in time interval Δ = 7, which is obtained by numerically integrating differential equations (8) and (9).