Cyanobacterial clock, a stable phase oscillator with negligible intercellular coupling

Abstract

Accuracy in cellular function has to be achieved despite random fluctuations (noise) in the concentrations of different molecular constituents inside and outside the cell. The circadian oscillator in cyanobacteria is an example of resilience to noise. This resilience could be either the consequence of intercellular communication or the intrinsic property of the built-in biochemical network. Here we investigate the intercellular coupling hypothesis. A short theoretical depiction of interacting noisy phase oscillators, confirmed by numerical simulations, allows us to discriminate the effect of coupling from noise. Experimentally, by studying the phase of concurrent populations of different initial phases, we evaluate a very small upper limit of the intercellular coupling strength. In addition, in situ entrainment experiments confirm our ability to detect a coupling of the circadian oscillator to an external force and to describe explicitly the dynamic change of the mean phase. We demonstrate, therefore, that the cyanobacterial clock stability is a built-in property as the intercellular coupling effect is negligible.

Fig. 1.

Comparison between the solution of Eqs. 4 (solid line) and numerical simulations of 10,500 interacting phase oscillators. The mean phases μ_M(εt) and μ_m(εt) (a), order parameter of the majority population ρ_M(εt) (b), and order parameter of the minority population ρ_m(εt) (c) are represented for three values of D/ε, 0.48 (red line), 0.96 (green line), and 1.44 (blue line). Three simulations are superimposed for each value of D/ε, with different values of the coupling constant: ε = 0.2 (dashed line), 0.1 (dash-dot line), and 0.05 (dotted line). (a Inset) Similar variation of μ_m(εt), obtained from Eqs. 4 for D/ε varying from 0.01 to 10.

Fig. 2.

The mean phase of the minority population is unaffected by the presence of a different phase majority. (a) Recorded bioluminescence of individual wells. Red and black lines indicate wells containing only luminescent cells previously entrained at opposite phases (A and C, respectively). Orange, blue, brown, and green lines represent wells containing mixtures of a luminescent minority with a 20 times larger population of WT cells as follows: (a, A), (a, B), (a, C), and (a, D). Lowercase denotes minority and uppercase indicates majority mean phase. The mixtures have a bioluminescence level initially 20 times and finally 30 times lower than wells containing only the luminescent strain. Each experimental condition is represented by 8–12 independent wells. (Inset) The four phases of entrainment are separated by ≅π/2 rad, with A leading B, in opposition to C and lagging D. (b and c) The circadian oscillation s(t) for the luminescent strain alone (b) and for the mixtures (c), represented with the same colors as in a. s(t) = i(t)/i(t) − 1, with i(t) the well bioluminescence and i(t) its baseline obtained by a fast Fourier transform (FFT) smoothing of i(t). The oscillations of A and C still have opposite phases at the end of the experiment. For the mixtures, the oscillation of the minorities appears unbiased by the presence of different phase majorities. (d) Instantaneous mean phase of the minorities μ_m(t) extracted from s(t), represented with the same colors as in a.

Fig. 3.

The average minority phase <μ_m(t)> for three initial phase differences μ_m⁰ − μ_m⁰: = −π/2, π, π/2 and the reference 0, represented, respectively, by blue, brown, green, and orange lines (error bars are the standard error of the mean). For each phase, the average is taken over ≅80 individual wells. Black lines show 95% lower and upper confidence limits of the linked linear fit of the ±π/2 curves.

Fig. 4.

Cyanobacterian oscillators respond to an external force as phase oscillators. (a) Entrainment lighting E(t) is applied for ≅14 days and then kept constant for the next 3 days to confirm the phase change. The light oscillates around an average of 500 lux, with a relative amplitude c = 0.4 and a 24-h period. Two of the four entrainment phases, separated by π/2, are shown. The gray vertical line highlights the transition to constant lighting. (b) The phase of the related entrained oscillators represented in a ω-rotating frame. The orange, green, blue, and brown symbols and error bars correspond to the experimental average phase <μ(t)> − μ₀ and standard error of the mean; the matching colored lines are the fits from the Adler equation with ε the only free parameter. Dashed black lines are the numerical simulations of Eq. 6, with the same ε. (Inset) The coupling constant ε increases with the relative amplitude of entrainment c.