Driving Potential and Noise Level Determine the Synchronization State of Hydrodynamically Coupled Oscillators

Motile cilia are highly conserved structures in the evolution of organisms, generating the transport of fluid by periodic beating, through remarkably organized behavior in space and time. It is not known how these spatiotemporal patterns emerge and what sets their properties. Individual cilia are nonequilibrium systems with many degrees of freedom. However, their description can be represented by simpler effective force laws that drive oscillations, and paralleled with nonlinear phase oscillators studied in physics. Here a synthetic model of two phase oscillators, where colloidal particles are driven by optical traps, proves the role of the average force profile in establishing the type and strength of synchronization. We find that highly curved potentials are required for synchronization in the presence of noise. The applicability of this approach to biological data is also illustrated by successfully mapping the behavior of cilia in the alga Chlamydomonas onto the coarse-grained model.

Figure 1

(a) Selected frames showing a single bead driven by optical traps through the geometric switch rule. The direction of the gradient of laser intensity is shown by the arrows. The direction of the gradient is reversed when the particle reaches a position defined by the dashed line; the bead is maintained in oscillation. (b) Mean driving potentials for a bead driven in potentials of positive (1), null (2), and negative (3) curvature. (c) Schematic showing that a generic potential is modeled in the theory by “two slopes.” The curvature parameter $c$ is measured from the mean potential in (b) by fitting the slopes at $-A/2$ and $A/2$, giving $F_b$ and $F_e$ for both decreasing and increasing potentials. Experimental parameters in (b): $A=3.1\mu \mathrm{m}$, $d=10\mu \mathrm{m}$, $T=296\mathrm{K}$, $F_0\sim 0.53\mathrm{pN}$, and $c=0.38(1)$, $-0.11(2)$, and $-0.40(3)$. Experiments are 2 min (single bead) and 5 min (two beads) long, while simulations in Figs. 3, 4 are equivalent to 2000 s.

Figure 2

Experimental tracks of two beads coupled hydrodynamically, for the three driving potentials characterized in Fig. 1b. Positive curvature leads to antiphase (AP) synchronization while negative curvature leads to in-phase (P) synchronization. In the limit case of linear potentials (zero curvature; central section of the plot), the beads do not synchronize.

Figure 3

(a) The phase/antiphase transition in the oscillation dynamics of the two particles is observed in experiments (●), and simulations (open markers), and is adequately captured by theory (lines). Conditions are $A=3.1\mu \mathrm{m}$ and $F_0\sim 0.53\mathrm{pN}$, and noise strength $\xi =2k_{B}T/(A{F}_0)$ is varied by changing the temperature [from (+) to (⋄): $\xi \approx 1.68\times {10}^{-7}$, $8.38\times {10}^{-4}$, $4.96\times {10}^{-3}$ (room temperature), $1.68\times {10}^{-2}$, and $8.38\times {10}^{-2}$]. $⟨Q⟩$ characterizes the state of oscillations (1 in antiphase and $-1$ in in-phase oscillations). Experiments should be compared with the simulations in (●). At any temperature, the state of synchronization is either in phase or in antiphase, depending on the sign the curvature $c$, with a sharper transition at lower temperature. The transition is smoother when noise is added. Theory fits very well with experiments and simulations up to room temperature. Inset: phase diagram showing synchronized states in phase (P), in antiphase (AP), or not synchronized (NS). Theory (lines), simulations (+), and experiments (○) show, for each value of the noise strength $\xi$, the value of $c$ for which $|Q|=1/2$. The dotted line shows the noise strength estimated for Chlamydomonas. The probability distributions of $Q$ at $T=296\mathrm{K}$ in experiments (b), simulations (c), and from the theoretical argument (d) show that in the presence of noise, the peaks are still centered around $\pm 1$ but with a large spreading. The theoretical curves are interrupted close to the zero crossing, since the analytical formula fails to describe the state when synchronization is very weak or lost (see the Supplemental Material [22]). In the simulations, $\varsigma =1\mu \mathrm{m}$. The synchronization of two beads is relevant to understanding the biflagellated organism Chlamydomonas, whose flagella perform periodic orbits in plane. The forces exerted on the fluid can be modeled as arising from a point source (a bead) and decomposed into orthogonal directions, as shown in panels (e) and (f); the consequences for synchronization are discussed in the text.

Figure 4

Autocorrelation of the phase $t_1$ between the two beads in simulations for different values of $c$: $-0.42$ (×), $-0.20$ (□), $-0.0071$ (⋄), 0.20 (▵), and 0.42 (▿) fitted to exponential decays (solid lines). Inset: simulated decay coefficients $\kappa$ for $\xi =4.96\times {10}^{-3}$ (+) and $\xi =8.38\times {10}^{-4}$ (×) are in good agreement with the theoretical values (line). The strength of synchronization is the strongest for the highest curvatures (high $|c|$). Theory and simulations differ for $c\approx 0$ because the assumption that oscillations are nearly in phase or in antiphase becomes wrong; when the noise strength increases this happens for larger values of $c$. Parameters are the same as in Fig. 3.