Dynamical quorum sensing: Population density encoded in cellular dynamics

Abstract

Mutual synchronization by exchange of chemicals is a mechanism for the emergence of collective dynamics in cellular populations. General theories exist on the transition to coherence, but no quantitative, experimental demonstration has been given. Here, we present a modeling and experimental analysis of cell-density-dependent glycolytic oscillations in yeast. We study the disappearance of oscillations at low cell density and show that this phenomenon occurs synchronously in all cells and not by desynchronization, as previously expected. This study identifies a general scenario for the emergence of collective cellular oscillations and suggests a quorum-sensing mechanism by which the cell density information is encoded in the intracellular dynamical state.

Fig. 1.

Two possible explanations for the lack of collective oscillations at low cell density: simulations of populations with random initial conditions. The gray lines represent the evolution in time of four oscillators within a population (n = 100), and the black line represents the macroscopic observable, the average over the population. (a) Incoherence. Cells progressively loose their mutual entrainment, and their average is asymptotically stationary up to finite-size fluctuations. (b) Dynamic quorum sensing. Cells have a coherent motion and stop oscillating in synchrony with the medium.

Fig. 2.

Amplitude A of the collective glycolytic oscillations as a function of cell density. The open and filled circles correspond to sustained and damped oscillations, respectively, of the NAD(P)H fluorescence relative to the mean fluorescence signal. Cell density is reported as dry weight (dw). The continuous line is a fit with the reduced model (Eq. 5 in Mathematical Model). (Inset) The predicted linear relation between A² and 1/dw. The vertical error bars indicate the maximum and minimum values observed, and the horizontal error bars indicate two independent determinations of cell density. See SI Appendix 1 for data analysis.

Fig. 3.

Angular frequency ω of the self-sustained (open circles) or damped (filled circles) collective glycolytic oscillations as a function of cell density measured as dry weight (dw). The continuous line is a fit with the reduced model (Eq. 3 in Mathematical Model). (Inset) The predicted linear relation between 1/ω and 1/dw. The error bars are as in Fig. 2. See SI Appendix 1 for data analysis.

Fig. 4.

Model of the intracellular oscillator used to simulate the individual cells of Eq. 1. The plane of oscillations (shaded gray) is spanned by the complex eigenvectors of the origin, associated with the eigenvalues λ₀ ± iω₀. The stable perpendicular mode is associated with a third, stable eigenvalue ∣λ_fast∣ ≫ ∣λ₀∣, quantifying the rate of relaxation toward the plane of oscillations. The coupling to the external medium takes place along the direction of the diffusing species (Aca). This direction forms an angle θ with the plane of intracellular oscillations.

Fig. 5.

Numerical simulations of Eq. 1 for a population of n = 100 limit-cycle oscillators. The parameters are derived from the experimental data or chosen according to the hypothesis of time-scale separation and fast diffusion. Amplitude A (a) and frequency ω (b) are plotted against cell density. The experimental data of Figs. 2 and 3 (black circles) and the reduced system (solid line) are compared with simulations of a population of identical oscillators (gray dots) and of a population with a mismatch in the frequencies of the individual oscillators (Gaussian distribution with a relative standard deviation of 15%; black dots). The parameter values are ω₀ = 0.17 s⁻¹, λ₀ = 0.015 s⁻¹, τ = 0.16 s⁻¹, g = −3.8 s⁻¹, θ = 87°, λ_fast = −500 s⁻¹, and d_aca = 300 s⁻¹. See also Fig. 4 and Mathematical Model.